Well, but what does “caring” mean? It means that their well-being affects your emotions. At its very core, you wanting to help people you care about comes from wanting to create positive emotions in yourself or avoiding negative ones (possibly in the future, it doesn’t have to be an immediate effect). If those emotions weren’t there, you wouldn’t actually care and thus not do it.

Edit to clarify: I’m not being cynical or pessimistic here, or implying that this means that everyone is egotistical because of this. The point I was trying to make is that defining egotism vs. Altruism is a little bit more complex than just looking at whether there’s something in it for the acting person. We actually need to look at what’s in it for the acting person.