Does human life have innate value over that of other animals? If so, why? And is it wrong to murder another human, but morally permissible to hunt or fish?
One aspect (not the only valid aspect, but the one I'll be covering here) of looking at this is from the social contract angle. In essence, since we live in communities together with other humans (and have done so as long as homo sapiens exist) it makes sense to agree to avoid killing each other in most circumstances (e.g. capital punishment, or religious sacrifices, or war) - defining any other killing as murder, and defining murder as Very Bad. It's just a very practical thing to do; if we establish social norms that I won't kill you and you don't kill me, then we're usually both better off than before. And at least part of the murder prohibition certainly is based on this aspect.
While the boundaries of social contract are in general arbitrary - there certainly can be social contract that highly values the lives and rights of one "tribe" (for a very, very wide interpretation of "tribe") and disregards the lives and rights of everyone else, there's still a major difference regarding animals; it makes sense to extend the social contract to all other people surrounding you (including all other "tribes"), because you'd want them to follow that social contract regarding yourself. It's not automatic (there are plenty nasty examples in history, including quite recent history), but in general that's a solid trend. With animals, on the other hand, such social contract isn't possible, because it can't be mutual, there's no reciprocity, they aren't members of our society and can't be because they lack the capacity to do so.
Our relationship with animals is inherently asymetrical. Bears can't peacefully join our society and obey its rules even if we'd grant them full human rights, and salmon won't be productive members of our society if they'd have equal rights, nor would they pose some threat to us if we'd not agree to their demands, so it's not in our interests to treat them as if they had equal rights and value - contrasting this with all the different groups of people whose lives had (de facto) less value in various societies just a few centuries ago, and the way how these differences have faded - mostly because all those people actually are (or can be) valuable members of societies and it's good for me and everyone else if they're included in our social contract.
You have the situation entirely backwards. It's not that we give humans special privileges but that we don't try to engineer the interactions of animals the same way we try to engineer the interactions of humans. That is, we exempt animals from having to comply with human-created rules because of humility and basic sanity.
Think about what we do when a bear eats a human. Now imagine if we tried to do the same thing if a bear ate a salmon or a deer. We would wind up taking complete human dominion over bear society in a way that doesn't make any sense at all.
If a bear can eat a salmon with impunity, why can't a human? Are humans specially of lower worth and entitlement to eat salmon than bears? Where would that come from? Why would humans be an exception here?
The right question is where would we get the right to force the societies of other animals to play by human rules that don't make any sense in that context? And why should humans be uniquely prohibited from benefiting from other species?
IMO your question addresses an important ethical problem. First one can ask:
Does human life have indeed innate value over that of other animals?
In my opinion, the answer is no. Human life does not have such superior value. Because values do not exist in nature. Instead they result from our decision to attach respect and esteem to certain objects or to a certain behaviour or attitude.
Of ourse, in ethics this decision often results from our innate feelings or more precisely: from the empathy we feel for humans or animals similar or close to ourself. We feel empathy with the members of our family, but we do not feel empathy with an insect.
As a consequence, in recent years a movement developed to generalize our human centered ethics to a system of ethics which also includes animals. A proponent of this discipline is Peter Singer. For a first information see https://en.wikipedia.org/wiki/Peter_Singer
Innate or intrinsic value is a kind of value such that when it is possessed by something, it is possessed by it solely in virtue of its innate or intrinsic properties. (Ben Bradley, 'Two Concepts of Intrinsic Value', Ethical Theory and Moral Practice, Vol. 9, No. 2 (Apr., 2006), pp. 111-130: 112.)
To say that human beings have innate or intrinsic value, then, is to hold that all humans and only humans, and therefore no non-human animals, possess the relevant properties.
It is reasonable to ask (a) what these properties are, (b) what is it about them that confers innate or intrinsic value, and (c) what axiological epistemology is available to us to determine the existence and nature of such properties.
I'm not aware of any theory that delivers credible candidates for (a) - (c).
It is one thing intrinsically to value all and only human beings - any form of subjectivism can do that. This is merely an attitudinal matter. It is quite another to say that objectively all and only human beings have innate or intrinsic value. That is a truth-claim and I don't see how it can be made good. I am open to rational suasion, however.
Religion and philosophy are of course not the same thing, but there is a lot that overlaps, and given that for a lot of people in the world, religion has a profound effect on their personal philosophy, I think this answer is important, and deserves to be listed among the other answers.
I won't go on and on defending it philosophically, as it rests on belief in religion in general, but it really deserves to at least be mentioned and understood.
Christianity and Judaism (and certain sects of Islam?) teach that human beings were created in the Image of God, the imago dei. If this doctrine is true, then it follows directly that there is an intrinsic value placed on human beings.
This is a very complex theological doctrine that pervades how we think about and treat other human beings. It is very much ingrained in Western thinking. It has historically been a motivation for much social change.
According to this view, animals, however, were not created in the Image of God, and as such, do not have the same intrinsic value as human beings. It's that simple.
Note however, that this does not mean that mistreatment of animals is morally okay according to religious doctrine. While not bearing the Image of God, they are part of God's creation that humans are supposed to look after and care for.
Also note that this view is not restricted to people who hold a literal interpretation of Genesis 1. Many believers of the Torah are happy to take Genesis 1 as theological/moral teaching, rather than as scientific/historical teaching. This still leaves many important doctrines to be gleaned from Scripture, including the doctrine of the Image of God. This begs the question, if humans evolved from animals in ages past, when did the Image of God enter the picture? Who knows.
Why does human life have innate value over that of other animals?
In my opinion, you just answered your own question. Observe:
If you want the why of it answered in a more detailed and analytical sense, neuroscience research for xenophobic aggression in mammals or fauna in general is probably your best avenue. Here's a paper that reviewed it for Naked Mole Rats: https://www.sciencedirect.com/science/article/pii/S0003347296902997
Humans are, in a way, a special species. We ask questions that go far beyond the need to get by and survive which apparently suffices to animals and plants. For example, we ask: Why are we here? What is our purpose? By asking and analyzing we get insight into nature and end up being able to do what every animal can and more - albeit with the help of self-built devices.
That indicates we are both part of nature and apart from it, standing in a position to observe and analyze. That suggests we have a value higher than the rest of the world around us and the decision to consider it wrong to kill a human flows from that though not everybody makes that decision (at least not fundamentally.)
However, as we are also part of nature, we still have to understand and respect our role within it. Everything plays a role and has innate value just by the virtue of existing. Using our environment in harmony with the cosmic laws that define these roles is important, otherwise we create damage to the entire system.
Quoting religion (the Bible in this case): Humans are masters of creation, created in the image of the creator. That means our ability to observe, analyze and create is a divine aspect. So, in a way, killing a human is like killing the creator.
Still, while this sets us apart from nature, allowing us to transcend it, a part of us belongs to it and we need to understand how we are fitting into the entire context.
Why does human life have innate value over that of other animals? Why is it wrong to murder another human, but morally permissible to hunt or fish?
If I love myself, then I love all others as well. Other people, of course. But also animals, trees, flowers, planets, stars, and ultimately the whole universe.
Yet as a biological organism, my body uses energy and quickly becomes depleted of nutrients, which I must replace by consuming foods such as plants, fish, and animals. I don't enjoy killing them, so I am very careful not to be wasteful. I'd kill them as humanely as possible. I try to get by on just eggs and milk. I eat vegetarian dishes, often. And I even go so far as to apologize to them, and to express gratitude to them, for using them as food.
Furthermore, I wouldn't even consider cannibalism, or eating another human being. For me, that is strictly taboo. Why?
Well of course I already mentioned that I love people. But I also sincerely admitted to loving all creatures. That's why I don't believe in exploiting animals. I only kill and eat what I absolutely must have to maintain healthy life.
But I do eat animal meat, while I would starve myself to death before engaging in cannibalism (even if they had died of natural causes or accident). I would kill a game animal to eat, but would feel extreme guilt and shame over killing a human. The thought of eating human flesh is so revolting I wouldn't be able to get or keep it in my stomach. I might be forgiven for killing an animal for food, yet intentionally killing a human would be unforgivable.
So, what is the distinction between loving humans and loving [other] animals?
What is the distinction between the value of humans vs. the value of animals? And why is it innate?
The “depth” of love is to be explained in terms of a notion of identification...in identifying with one’s beloved, one might have a concern for one’s beloved that is analogous to one’s concern for oneself.
When I am hungry, before eating I evaluate my food by answering a few questions about its quality. The first question usually is, "What species?" Fortunately, I've never been offered human flesh for dinner, but I wouldn't eat it anyway, because I personally identify with the human species. Perhaps cannibals do not, at least not enough to inhibit them from doing so.
Love, c2017, Stanford Encyclopedia of Philosophy
First, because humans write the rules we use, and we're not entirely disinterested. It would be awkward to make it illegal to kill other animals in general, but we're generally better off treating the killing of other humans as a generally bad thing.
Second, we maintain a moral boundary by treating humans as moral actors, but not other animals. We hold humans responsible for their own actions, but treat actions of other animals as acts of nature. We're then saying that killing a moral actor is different from killing something natural without moral sense.
With reference to Geoffrey Thomas' excellent answer, I propose a narrowing:
It would be difficult indeed to make a universal claim that humans have intrinsic or innate value. But I believe that this is unnecessary; after all, are we attempting to find a bear or jaguar guilty for killing a human?
Instead, I would say that humans and only humans have innate value, and that must be respected by all humans. But other animals that harm humans do not become guilty for their actions, and so in the same way humans are not guilty for killing animals*.
*It would still be a matter of guilt if the animal-slaughter was done contrary to other norms (such as the norms against animal abuse).
I'm astonished that no answer seems to directly address the core issue: Humans are sentient beings. This is what distinguishes us from (other) animals, and what makes us valuable in each other's eyes.
I'm aware that "sentience" is a fuzzy concept and that it is not entirely sufficient to explain the value we bestow in fellow humans even when their mental capacity is minimal (with few exceptions, like Peter Singer). But I believe it's the underlying criteria which lets us establish a hierarchy of values in the animate world. There is a pretty clear hierarchy we establish in animals: Big apes and dolphins are more valuable than earth worms, and if we ever made contact with sentient aliens they would be on the same level as humans on this virtual "value scale".