The meaning behind that is a little obscure, and it could go in different directions depending on how you look at it. To me, it means that it's not what you know or don't know that makes you stupid, but what you do with that knowledge. Being stupid is an action, and a "stupid person" is a person who makes stupid actions. The term "as" is a little hard to explain, but I like to think of it as like an equal sign. In this example stupid = what stupid does. Take like the term "It's as good as new." The statement basically applies the quality of "new" to "good", then applies that version of good to whatever "it" is.

It comes from an old saying "Handsome is as handsome does" or "Beauty is as beauty does." (meaning: if you do good/beautiful things, you are a good/beautiful person) "Stupid is as stupid does" is from Forrest Gump. He has a very low IQ and some say he is "stupid." According to this saying, he is not stupid because he makes smart choices and lives a very nice life. "as" here is used to form a conjunction, and means "because" or just an equal sign like Mytch said, so ... "stupid is = stupid does" or "one is stupid because one acts stupid."

you do stupid stuff because you are with a stupid person