Biff steals things because it is his passive-aggressive way of getting back at people he feels have unfairly done better in life than he has, but also because stealing gives him an excuse to quit whatever dead-end job he happens to be in at the moment.
Consider Act 1, in which Biff and Happy are speaking alone.
Biff: Well, I spent six or seven years after high school trying to work myself up. Shipping clerk, salesman, business of one kind or another. And it's a measly manner of existence...And always you have to get ahead of the next fella. And still -- that's how you build a future.
A little later in the scene, Biff wonders about a former employer: "I wonder if Oliver still think I stole that carton of basketballs." Happy consoles, "Oh, he probably forgot that long ago. Anyway, he didn't really fire you." But Biff replies, "Well, I think he was going to. I think that's why I quit."
This pattern of subtle revenge and avoiding of mundane responsibility is exposed to be a nearly life-long pattern for Biff, and one that will continue.