Maybe a better question is,
"Why do we have this crazy rule?".
The genesis of the rule is akin to the same reason we have the infield fly rule: To prevent the defense from exploiting a loophole in the rules to gain an unintended advantage. The uncaught third strike rule dates back to the earliest days of baseball. It survives today to confuse and confound coaches, players, parents, and, unfortunately, even some umpires.
Everybody knows three strikes and you're out. But what the rule actually says is that the batter is out when the third strike is caught. "Caught" in this case means that the ball is airborne, or in-flight, from the point of the pitcher's release to the catcher catching it. If the ball hits the ground, the batter, the umpire, or anything else besides a defensive player, it's no longer in-flight.
The rules also state that when the third strike is not caught the batter becomes a batter-runner, same as if the ball was batted into play, and the batter-runner may advance to first base. Only sometimes...they can't! Here's why we have the exception in the rules for when they can't.
In baseball's early days this exception wasn't in place. The batter could always run if the third strike wasn't caught. Catchers soon found a way to exploit that rule. If at least first base was occupied (ie: there were force outs to be had on the bases), a crafty catcher would let the ball hit the mitt then purposely drop it. Now the batter had to run. Since runners on base would stay close to their base on a normal strike pitch, all the catcher had to do was pick up the ball, throw to second or third (wherever you had force outs) and the defense could easily turn two or three easy outs on a play that should really just be one out.
Somewhere around the turn of the century, the rulesmakers decided to close that loophole. So they added the exceptions for when a batter
doesn't become a runner.
With 0 or 1 out and first base occupied (at the time of the pitch) the batter may not run. Since the batter doesn't become a batter-runner, there are no force outs to be had. This removes the defense's ability to get multiple outs and that removes the incentive for the catcher to purposely drop the ball. In that way, this rule serves the same purpose as the infield fly rule.
With 2 outs, the defense only needs one out to finish the inning. Purposely dropping the third strike with 2 outs would be...stupid. It offers no advantage and, in fact, would disadvantage the defense! So, when there are 2 outs, the rules still require the defense to complete the play, either by catching the pitch or making a throw and tag at a base.