In this After Effects tutorial, Sam Bendix covers an introduction to the basics of the puppet tool.
Give you a quick tutorial on the puppet tool. So you're going to want to select the layer you want to use the puppet tool on. I'm going to use it on my cattail to make it move. You're going to want to go up to the push pin icon and if you hover over your object you can put a push pin anywhere in the yellow area. And so you'll want to put a push pin where you want something to move or where you don't want something to move. In this case, I want my cattail to move in three spots and I don't want it to move everywhere else so I'm going to put anchor points around. And honestly, there's probably a better way to animate this, that I don't need all these anchor points, but I don't know.
So you can see down here that we have a new little tab, it's called mesh. And in the mesh deform, that's where all your puppet tool, your pins are, and then you can alter the placement and stuff in this menu. Or you can come up and you can hover over one of the pins you placed and it'll give you a move prompt by the mouse and so you can just simply grab one of the pins and move it that way, like that. And it will stretch your shape layer or file layer, so just be conscious of that. So, yeah, this is the basics of the puppet tool. I hope it helps.