Obviously your mileage may vary, but I find it helps to imagine a stranger as someone else's family/friend. If I think of how much I care about people close to me, and imagine that that stranger has people who care about them as much as I can about my brother, then I find it easier to do things to help that person.

I guess you could say I don't really care about them, but care about the feelings of caring other people have towards them.

If that doesn't work, this is how I originally though of it. If a stranger passed by me on the street and collapsed, I would care about their well being (I know this empirically). I know nothing about them, I only care about them due to proximity. It offends me rationally that my sense of caring is utter dependent on something as stupid as proximity, so I simply create a rule that says "If I would care about this person if they were here, I have to act like I care if they are somewhere else". Thus, utilitarianism (or something like it).

It's worth noting that another, equally valid rule would be "If I wouldn't care about someone if they were far away, there's no reason to care about them when they happen to be right here". I don't like that rule as much, but it does resolve what I see as an inconsistency.