"He is better than _____." (1) I (2) I am?
They are both grammatical, the first being a truncated form of the second. In both, than functions as a conjunction. However, He is better than I sounds hopelessly formal, at least in British English. The usual form is the equally grammatical He is better than me, where than functions as a preposition. When a preposition is followed by a pronoun, the pronoun is in the accusative case.