You have two choices, no make that three. If you want to stay inside AE then you'll have to either keyframe all of the stickmen falling down by hand carefully estimating when they physically touch each other or purchase a script called Newton. The extrusion can be done inside AE using Ray-traced rendering options for each layer but the render will be slow if you have a CUDA compatible NVIDIA gnu or excruciatingly slow if you don't. You could also do a fake extrusion using copies each stickman offset by 1 pixel in Z space and parented together. If you can live with flat stickmen then this will be a lot easier to do in AE and take a lot less time to render.
By far the easiest way to do this effect is with a 3D app because most of them have things like collision detection and physics engines. Blender is open source and Free and could do this in a heartbeat. There are lots of learning resources. C4D Lite, which comes with the latest versions of AE is also a possibility, but not as easy to use because it is the light version and the physics plug-ins are not as robust.
Here's one tip, for both AE and 3D apps. The anchor point, which is the center of rotation, needs to be set at the feet of your stickmen for the animation to work.
Blender looks amazing ! But I don't have a ton of time to learn such a complex program.
This is the base vector I want to expand....
I have Photoshop and After Effects (with cuda + compatible GPU) and now also blender, how do I make that stickman 3d ? I think the first step is to generate that 3d object from the above vector file, then create a floor, place my stickmen and learn how to move the cameras, and change the color of a stickman.
Thats my job for today. I'll figure out the falling part later, but thats not nearly as important actually.
Where to begin ? These programs are so complex.