Why does Unix store timestamps in a signed integer?

Early versions of C didn't have unsigned integers. (Some programmers used pointers when they needed unsigned arithmetic.) I don't know which came first, the time() function or unsigned types, but I suspect the representation was established before unsigned types were universally available. And 2038 was far enough in the future that it probably wasn't worth worrying about. I doubt that many people thought Unix would still exist by then.

Another advantage of a signed time_t is that extending it to 64 bits (which is already happening on some systems) lets you represent times several hundred billion years into the future without losing the ability to represent times before 1970. (That's why I oppose switching to a 32-bit unsigned time_t; we have enough time to transition to 64 bits.)

It's to support timestamps and dates before January 1st, 1970.