When you spend a lot of your free time reading tea leaves, you occupy yourself with that pastime, meaning it takes up your time and keeps you busy.
An army can invade another country and occupy its territory, and this use of the word reflects its original Latin meaning, which was "to seize." There are also less aggressive forms of occupation, as with the family of chipmunks who occupy your garden. If you are the mayor of your town, you occupy an important office. And if you're a fan of Andy Warhol, a series of Campbell's soup can paintings may occupy a prominent place on your walls.