。
I said I would make her happy.
Literal
I [topic-は] her [object-を] happy [make-にする] [quotative-と] said.
僕 marks the speaker as male but softer/more polite than 俺. 幸せにする is the stock phrase 'make (someone) happy' using the X + に + する 'make into X' pattern with 幸せ 'happiness' as the target state. と言った uses the quotative と to embed the promise as reported speech. A classic romantic or wedding-vow phrasing.