We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going:
http://dictionary.cambridge.org/grammar/british-grammar/in-into

Example by @Fard:   

* I put my hands in my pockets to keep them warm.  
* Slowly, he put his hand into his pocket and snuck out a folding knife.

In "Put your hands into your pockets," we're talking about the movement of putting your hands.