"In" emphasizes the situation of something or someone being inside of something. For example: I am in the library. The cheese is in the refrigerator. There is a mouse in my house. Into emphasizes the action of a thing or a person entering or being put inside something. For example: I am walking into the library. He put the cheese into the refrigerator. I saw the mouse creeping into my house.

In addition to Jura's explanation, "get in" and "get into" are phrasal verbs. For example, "she got into politics" means she became involved in some issue. To be elected to some office or achieve entry to an exclusive school or club, one is said to have "gotten in".