In American football, what do the numbers that the quarterback calls out before a play mean?

Just that...he's calling the play. The quarterback should stay away from the huddle until it has been formed. Then he should get in, call the play, and get the momentum going. The main thing to remember in calling plays in the huddle is to keep your head up and look directly at your players. Some quarterbacks get their heads down and wind up talking to the ground. If you get your head up, look a man right in the eye, and tell him, "Brother, I want you to block," there's a pretty good chance he'll get the job done. Nobody likes to talk with a person who avoids looking at him, and this same psychology holds true for the relationship between a quarterback and his teammates.

https://www.espn.com/blog/carolina-panthers/post/_/id/30853/cam-newton-not-alone-saying-calling-a-play-shouldnt-be-a-paragraph

Who gets the football and where it is going

Mmmmm still not sure. Zbeck, your reference was helpful in that I now understand the importance of the cadence and tempo. But apart from that, if the play has been discussed in the huddle, why is it necessary to call it out again? Doesn't it let the other team know what's going to happen? Since my experience of American football doesn't extend much beyond Playstation, what (for example) does Blue 32 (as I heard Clark Kent call out in Smallville) mean and again, why let the other team know?

Exactly that! Play designation.

The quarterback usually calls out the signal for the ball to be snapped and may optionally change the play at the line of scrimmage before the ball is snapped by shouting out additional signals to the other players on his team. This is known as calling an audible.If an audible were to be called, it would be shouted somewhere between "Down" and "Hut". A common way to call an audible is to call out a color and number. One color is designated by the offense to signal a true audible, known as the "live color", and the number tells which play will be run. An example is "Blue, 42". If blue were the designated color, play number 42 would be run. If blue were not the designated color, no audible would be made and the play called in the huddle would be run.

https://en.wikipedia.org/wiki/Quarterback