If we take the World as absurd - without any meaning or end result, and believe in no such thing as God or afterlife (taking Camus' stance in Myth of Sisyphus), why should anyone, fundamentally, care about anything or anyone? Of course, in transactional relationships reciprocity is the norm, and therefore it makes sense to engage in transactional care (social/psychological needs). But is there anypoint in really caring about anyone? What arguments can we provide for caring in the face of absurdism?
One of the weaknesses of the Western philosophical tradition - something that Existentialism and Absurdism tried to address — is that Western philosophy discounts psychology (broadly put). It implicitly presumes that things must have reasons in the analytical, intellectual sense, and that leads to some unnatural conclusions. For instance, one might wonder why people bother to eat complex, carefully prepared, diverse meals. All one needs for continued existence is a bland mush of protein, carbohydrates, fats, and assorted nutrients, and one could save a heck of a lot of time and effort by (say) eating meal replacement bars from WalMart for every meal. But people don't; they like complex, diverse, 'rich & tasty' foods, and are willing to spend large portions of their lives seeking such out. Of course, someone could say that people seek out good food because it's pleasurable, but that's just a pseudo-rationale. 'Pleasure' is not an analytical concept; things are pleasurable because they are pleasurable, and for no other reason. It's a built-in bias of our biology, not a moral choice.
Caring for others, and being cared for in turn, is pleasurable. While we could get all of our basic social and personal needs met through simple transactional relationships, most people become bored with the shallowness of it, and seek out deeper and more complex relationships.
Thinkers like Camus and Sartre were focused on the 'objective' meaninglessness of existence almost to a fault; they failed to look at (at least as far as I recall off hand) how the act of making meaning out of life is intrinsically pleasurable. Your question boils down, I think, to a question of why we should make meaning out of life if life has no intrinsic meaning. And the only real answer to that is that we are happier when life has meaning.
Sartre proposes that caring is a reality of human psychology from which we cannot escape (along with psychoanalysis and a raft of feminists, but he is the biggest-name philosopher who put it clearly). In doing so he goes back to the Cynics.
This question is almost always asked this way around, by solipsists, by skeptics, by absurdists, by other forms of nihilists, etc. This is not the way it was originally proposed by this school.
Diogenes' first question was how (and why) to deface currency -- how to determine what not to care about. That is a much more useful question. But it is not whether to care it is when to care because we basically have no choice.
Humans as animals automatically care about far too many things: their next meal, the next generation, the opinions the others around them, how they might be remembered after death, whether they themselves might somehow survive death, how much to fear that, what death is... Piles and piles of automatically arising attachments. Since we are social creatures, we also project those concerns onto those around us and empathize with them. Babies symbolically get even with abstract shapes that treat other abstract shapes badly.
It is hard work not to care. Much harder work than choosing something to care about. Why do so? It is that impulse that one needs motivation for.
And how would you get the motivation to sustain that hard work unless you chose to care about it? If you take the world as absurd, why fight your biology that hard?
If 'the World is absurd - without any meaning or end result' then if caring is absurd, so is not-caring. 'Why should I care about anyone?' and 'Why shouldn't I care about someone - or everyone?' - both questions are themselves absurd as are any answers to them. 'Why should I care?' has no special salience; it's just itself one more absurd question among all the others. This is not my viewpoint but it appears to be a logical consequence of the position you define.
A further logical consequence is that my answering the question is an act of absurdity, but then so would be my worrying about this absurdity.
This text will be long, yet will give you one scientific way to look at it.
When reading, please turn off the emotions for a minute. Also lets look in a big scale, instead of looking individually - that said, lets try to answer "why should human species care?".
Then we can extract the individual answer for ourselves.
In the end, your question all boils down to the statement "nothing really matters" why should we care, right ?
And that could be interpreted as "we can live in any way we want, without considering or caring about others". But that's only one way of interpreting it.
To see the whole picture, lets go through all possible combinations of
not considering others in our actions. Then we will (try to) find a root cause, and then see where we are and where we go.
The possible combinations are:
We choose to die.
We choose to live, without considering others.
If everyone lives this way, chaos may begin and we all eventually die - just because someone else wanted it, didn't care or didn't consider others. Thus chances of dying are higher. It may contradict the choice itself - we still may die whereas we chose to live.
We choose to live, by considering others.
But "do whatever we want" gets partially limited when "considering others"... Thus, we give up a part of our "freedom of action" when considering others. Then we avoid doing actions that may harm others.
We do actions that help other, which in turn may also result into helping us.
The "caring" (which you mentioned) happens in this case too. We care about others too (childs, parents, etc), and get cared too.
If everyone lives this way, the chances that we will die are much less than in the first 2 cases.
Given all that, we can say that in the case (1) the chances of death are the highest, while in the case (3) chances of death are the lowest. Thus (3) is the most life-sustaining formula.
Keeping this in mind, lets look at the "evolution" part to see what evolution does with sustainable/unsustainable forms of life.
By evolution, all forms of life transform consistently. Unsustainable forms die out or transform into more sustainable forms. So, by evolution, we (humanity) are also naturally advancing from unsustainable (1 and 2) to more sustainable form (to the case 3).
Side note: "Considering others" (case 3) requires us to sacrifice a part of freedom, to prevent ourselves from harming others.
It explains why we eventually created rules, laws to limit ourselves.
They all fit into natural evolution - steps of transformation of humanity from less sustainable to more sustainable life species.
Where we are generally: Unfortunately, we can generally say that humanity is in the case 2. It still exists more than it should. To prove it - look around - wars, robberies and other acts of violence still happen every day.
Suicides also happen (1), yet they take less lives than wars/etc.
Also we can see actions fitting more into case 3, but we are far from saying that humanity mostly fits the case 3. We haven't advanced to it yet.
With all that information, we can say that:
To become a more advanced and sustainable species (civilization), we (humanity) need to care about others, consider others.
Natural evolution's laws brought us to this point. And if we don't advance more into it - the humanity extinction is inevitable and is matter of time (human self-destruction or cataclysms will happen eventually).
This theme may go longer and can jump into many other aspects; but I hope this text gave you enough points think (and perhaps answer your own question).
Right now I would sum this up (constructively) this way: we need to consider/care about others in order to become a more advanced life-sustaining (species) civilization, and lessen the chance of death/humanity extinction.
You may sum-up other ways - would be very nice to see them!
Thanks for reading!
About the interesting question in comments:
"Also, how would you explain, say, firemen, who risk their lives because they care? That certainly doesn't help them survive?"
Of course, that wouldn't help the firemen to survive. But it makes "humanity" more sustainable.
Take a look from evolution's point of view, which keeps life-sustaining formulas... and plays with whole species (not individuals).
Previously a human risked and saved more lifes than his own. That means this recipe helped to sustain life more, so this formula will be repeated, unless it becomes useless or starts to harm more lifes than it saves. Evolution works this way.
Of course you can add more personal reasons why firefighters safe lives... and those reasons will also be naturally part of the same evolution...
We care due to pragmatic reasons, not by altruism.
As individuals, our final goal is to survive (any other goal is subject to being alive). Our probabilities of survival increase if we survive as a group, so we care about the rest, about every member. Therefore, in practice, the survival of the individual is subject to the survival of the group (that's why from time to time we, as a society, could agree to kill someone or put others in jail).
So, if you want to survive, you need to care about the rest. If you want to reduce your probabilities of survival, you can stop caring about the rest. Teach your son to be an unpleasant person, to care about nobody, and the probabilities he'll end in jail or get dead will increase dramatically. The more sincere and caring you are about the rest, the more your probabilities of survival are.