I was watching a compilation video that included many clips in which a catcher, immediately after a pitch, threw to second base, and a fielder tagged someone out who was stealing from first. In many of those clips, you can see that fielder then toss the ball in the direction away from the pitcher, seemingly to the center fielder or the second-base umpire. Whom is he tossing it to, and why?
They’re tossing it to a teammate. It’s a common thing to do after completing a play; it’s sort of the baseball version of spiking the ball after a touchdown - but more respectful, not with the implication of belittling the opponent. You won’t see it when there are other players on base unless it’s the final out of the inning, or course.
I’ve always assumed that it was mostly a way to build team spirit - the out only really involved the catcher and the 2B or SS, but they throw it around the diamond to get the other guys in on the action. It’s also a good way to practice throwing - while these guys are very good, more muscle memory is always a good thing.
It is common for the ball to be tossed around the infield after any out before the inning is over and there are no runners on base, not just after a caught stealing.
The infielders' practice of throwing the ball to each other after recording an out, provided there are no runners on base. The purpose is as much traditional as anything, but it also serves to keep the infielders' throwing arms active.