It's a free and useful material to go along with a child's imagination (it can be a gun, it can be a sword, it can be a teacher's pointer, it can be a flagpole, it can be used as basis for all manner of constructions, you can use it to poke a things or make small holes, it lets you reach things which would otherwise be beyond your reach and so on...) plus its a complex enough object to be interesting in itself not just in shape but in texture, plus sticks come in lots of variations in shape and size.

Even the process of selecting the right stick for something you have in mind can be intellectually stimulating.

Or to put it in another way, sticks are a great basic elements for kids to invent and play their own games, they're usefull for a lot of things and you can find them for free on the ground.