Everyone looks down to first base while they wait to find out whether or not the batter swung.
Checked Swing
The pitch comes in, the batter flinches like they’re going to swing, but then they don’t swing all the way, and all of a sudden, the umpire and catcher are pointing at things. What’s happened is called a “check swing”: the batter saw a pitch, thought for a second that they wanted to swing, but at the last second, they decided to check themselves. The determination of whether or not they wrecked themselves is left to the umpires.
If the pitch that the batter checked on was a strike, then the umpire will call it a strike, much like if the batter just watched the pitch go by. Similarly, the umpire can decide for themselves whether or not the batter swung, and can call a swinging strike, accordingly. If the umpire thinks the batter did not swing, and that the pitch was a ball, then they will call a ball. If the catcher thinks that maybe the batter did swing, they can appeal the check-swing part of the call with the first-base umpire (for right-handed batters) or the third-base umpire (for left-handed batters). This is what the pointing is all about; the catcher appeals to the umpire in question, and the home-plate umpire points to indicate that they are granting the appeal. The base umpire then makes their determination, and play continues. The base umpire will indicate a swing with a kind of fist-pump motion (not dissimilar to the signal for “out”), or by waving their arms with their palms down (not dissimilar to the signal for “safe”). Note that batters cannot appeal a check swing call, though you will sometimes see them join in the pointing anyway.
Here’s the Fun Part: there is nothing in baseball’s official rulebook that defines what a swing is, nor anything to define what a check swing is. You might hear announcers talk about things like “breaking the plane,” in which the head of the bat might cross the front part of home plate, but nothing of that is in the actual rules. Which means that the umpire making the judgement is making a pure judgement call, which is always the sort of thing that goes over well, especially in tense situations where a call may mean the difference between extending a plate appearance, or striking out. No, whatever the umpire says, goes1.