If Jesus spoke Aramaic like the rest of the Jews in ancient times, where did Hebrew come from and why do random Israelis speak Hebrew?

Hebrew was spoken before Aramaic was adopted. Aramaic was the language of Babylon in the time of the Babylonian exile and was the main lingua franca in the Near-East. The Jews adopted it during the exile in Babylon and continued speaking it thereafter.

When Jesus was born Aramaic was the main vernacular language but Hebrew was still the language of religion. The Torah, and all the books of the Jewish Bible except most of the Book of Daniel, which is mostly in Aramaic, are in Hebrew and this was still used in prayer and study by literate Jews.

As for “random Israelis”- Hebrew is the language of İsrael so Israelis speak it in much the same way that “random Americans” speak English.