How come Brian is only one who understands Stewie on Family Guy

Same intellectual level?

Because they have a special form of communication.

Since they often go on adventures together, time traveling to the past and the future, it would seem to me that we see the world through Stewie's eyes and imagination. In truth we'd find out, Brian doesn't really speak Americanized English, but rather an ancient language the rest of the Griffin family speaks. To survive, Stewie pretends that it's actually English. Brian being a good dog, goes along with it.