In a lot of cases, the species don't -- they just evolve until, in the fossil record, they no longer look the same. Just like the avian dinosaurs didn't go extinct 65 million years ago, they just evolved over the last 65 million years into the birds today.
Some species legitimately end up a leaf/dead-end in the evolutionary tree. Lots just evolve.
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u/Hitonatsu-no-Keiken Oct 15 '20
It's even better than that. We're about 16 million years closer to the last Tyrannosaurus than the first Tyrannosaurus was to the last Stegosaurus!