I identify with, and care about, the person I will be tomorrow because we share a large fraction of our plans and sub-plans. The same goes, though to a lesser extent, about the person I will be one month from now, and so on.
I identify with the person I used to be yesterday because we share a lot of past experiences, and these experiences are resources I (and my future selves) may use to fulfill our plans. Even if I'm now very different from the kid who read Heinlein and Hofstadter, his experiences - my past - is what I draw upon. No one else has quite the same - I would dispute the assertion that "many years from now [a future me] will resemble [me] less than some actual people living today".
Some people do undergo large discontinuities; luck or catastrophy changes their plans to such a large extent that they have less in common with their past selves than others. Some people joke about that: "In a previous life I used to be a high powered executive, now I run a small restaurant." (More realistically, I could say of myself that in a previous life I was a programmer, and now I organize conferences and sell my services as a consultant.)
Even if my future changes dramatically, my past will still be the same, and my past will continue to supply me with resources for bringing about futures I desire. If my desires about the future change dramatically, I still need to weave a consistent story about "myself", in order to make effective use of those resources. The CxO's experience is repurposed as preparation for the running of a restaurant. His expensive health club, symbol of wealth, has become excellent preparation for a job that requires robust health. (In the less dramatic example, my project management experience turns out to be applicable to conferences and consulting work.)
What makes me me is this combination of a fixed past, and plans for the future.
From the dawn of civilization humans believed in eternal life. The flesh may rot, but the soul will be reborn. To save the soul from the potential adverse living conditions (e.g. hell), the body, being the transient and thus the less important part, was expected to make sacrifices. To accumulate the best possible karma, pleasures of the flesh had to be given up or at least heavily curtailed.
Naturally the wisdom of this trade-off was questioned by many skeptical minds. The idea of reincarnation may have a strong appeal to imagination, but in absence of any credible evidence the Occam’s razor mercilessly cuts it into pieces. Instead of sacrificing for the sake of the future incarnations, a rationalist should live for the present. But does he really?
Consider the “incarnations” of the same person at different ages. Upon reaching the age of self-awareness, the earlier “incarnations” start making sacrifices for the benefit of the later ones. Dreams of becoming an astronaut at 25 may prompt a child of nine to exercise or study instead of playing. Upon reaching the age of 25, the same child may take a job at the bank and start saving for the potential retirement. Of course, legally all these “incarnations” are just the same person. But beyond jurisprudence, what is it that makes you who you are at the age of nine, twenty five or seventy?
Over the years your body, tastes, goals and the whole worldview are likely to undergo dramatic change. The single thing which remains essentially constant through your entire life is your DNA sequence. Through natural selection, evolution has ensured that we preferentially empathize with those whose DNA sequence is most similar to our own, i.e. our children, siblings and, most importantly, ourselves. But, instinct excepted, is there a reason why a rational self-conscious being must obey a program implanted in us by the unconscious force of evolution? If you identify more with your mind (personality/views/goals/…) than with the DNA sequence, why should you care more for someone who, living many years from now will resemble you less than some actual people living today?
P.S. I am aware that the meaning of “self” was debated by philosophers for many years, but I am really curious about the personal answers of “ordinary” rationalists to this question.