To get a game started, all you have to do is start or open a conversation and send the basketball emoji. On Android, you can find the emoji under the "bell" tab in the emoji tray (next to the "send" arrow icon). On an iOS device, change to the emoji keyboard as you normally would, and you can find it under the "activity" tab (the soccer ball icon).
Once you've sent the emoji, just tap on it and the game will start. It is basically free throw practice. The ball sits along the bottom of your screen, and you swipe up toward the hoop to shoot.
Once you hit ten in a row, the backboard starts to oscillate about the screen to make it more difficult. The game is pretty basic, it just keeps track of how many shots you can hit in a row.
Messenger keeps track of the high score set in each conversation, so you and a friend can keep trying to one-up each other, if you're the competitive type.
Anything that happens beyond ten consecutive makes is a mystery to me, because that's as good as I could do; either the mini-game is kind of tough, or I suck at it.
A quick tip: It's easier to throw the ball straight if you use your finger, instead of your thumb.
That's pretty much all there is to it. If you get bored, you can always go back to playing Chessenger.