Let's start from the basics: property rights, aka the exclusive authority to determine how something you own is used.
A human owns himself, his life, his time, his productivity, the rest of his property, etc. He can decide what to do with his property, but others cannot without his consent.
And... that's all there is to it really: like most human rights, consent comes from property rights.
- You can break your own toys, but not mine unless I agree.
- You can stick things into your own bodily orifices, but doing so to someone else without their consent would be rape.
- You can do what you want with your life, but forcing someone to do what you want with their life would be slavery.
In all these cases, the fundamental part is who owns the thing.
Note there are so-called "human rights" which do not come from property rights, quite the opposite in fact... most noticeably, the generic "right to free stuff", which is a right to the work of slaves who will produce said free stuff. These are not human rights at all, but I'm digressing.
Anyway. Next we have reciprocity. A formulation could be "don't do to others what you wouldn't want done to yourself" or "The rules you wish to impose upon others must also apply to you." This is easily hackable when using detailed examples (for example a masochist could argue that a mandatory daily spanking for everyone would be totally fine since they'd like it a lot) so it's better to stick to general principles:
Reciprocity implies that: if you think property rights should not exist for other people, then they shouldn't exist for you either. This means you don't own yourself, which means you have no rights at all, and no way to argue why anyone shouldn't do anything they want to you.
Basically, the only alternative to the system of rights built upon property and reciprocity is might makes right.
After this... theoretical introduction...
We cannot see or feel consent.
Lack of consent can sometimes be physically measurable, usually by the amount of violence retribution that follows.
Why is it important when its existence cannot be proven?
Why should autonomy be respected?
Reciprocity: if you don't acknowledge the existence of other people's consent and autonomy, then yours does not exist either. This is why commies don't have rights, btw.
The importance of consent seems to rely on the principle of respect for autonomy or self-determination; for one to make their own decisions?
Yes, that's another way to say "property rights".
What if there is no ability to give consent? A child or unconscious patient
In this case the traditional approach is to assume the unconscious patient would express the same wishes as pretty much everyone else would if they could communicate. If a patient arrives in the ER bleeding and unconscious, it's pretty safe to assume they would agree with an attempt to save their life.
However, this is not as clear-cut as it seems: for example the patient may have beliefs that would lead them to prefer death to certain forms of treatment, or perhaps they actually wanted to kill themselves. Likewise the patient may be under the influence of drugs or other conditions that prevent the brain from working properly, thus unable to consent. In this case the solution is to leave clear directives beforehand like the "do not ressuscitate" directive so the patient's rights can be respected. However this is not foolproof and requires logistics.
Likewise, we can opt (or not) for an organ donor card, because... property rights over your organs.
Thus there is an important difference between a child and an unconscious patient: the latter can leave directives beforehand, and if they didn't, we can assume they were okay with the default way things are done.
But if consent can be implied and inferred from silence, why is it immoral to engage in sexual activities with a child or unconscious person?
Dude, come on.
First, "consent can be implied and inferred from silence" is not true, the person could remain silent because they're too afraid to talk, or other factors. "Silence is consent" would only be valid if the person was free and willing to speak without any consequences.
Second, it is not necessarily immoral to have sex with someone unconscious if that was agreed upon beforehand, although one must take extra precautions because they won't be able to communicate if they're hurt. And what if they're also tied up and gagged, or their mouth otherwise occupied with something? Sometimes it's complicated. For example I had this girlfriend who would pass out when coming too hard, but she had told me to keep going anyway because, quote "it got her off". I'm not the kind of man to say no to a lady's wishes, so I did exactly that, but I did keep a wet towel in an ice bucket to wake her up.
So yes, prior consent is a thing, how else would you wake up your GF with a kiss? Even worse, if she wakes you up with a cuddle, are you going to sue for inappropriate groping because the sleeping can't consent?