"I am happy because I am rich." Exactly which part does the subordinate clause modify?

I am happy [because I am rich].

The subordinate clause is "I am rich".

The bracketed element is a preposition phrase headed by the preposition "because", with the subordinate clause as its complement.

The preposition phrase is functioning as an adjunct of reason in clause structure, i.e. it's a modifier in (and thus part of) the VP "am happy because I am rich".