They get swirly eyes and fall over covered in soot Wile E. Coyote style. In some cases the Pokemon themselves don't even blow up (Clefairy in the Mt. Moon ep), but just make a big explosion in the general area. Then again, that was from Metronome so it might've been different.
I feel like there were a few early episodes of the anime that showed Voltorb exploding with no sign of them afterward. It's also telling that the animations for Selfdestruct and Explosion pre-Gen III had the Pokemon's sprite disappear after the explosions.