。
She looks really happy.
Literal
She [topic-は] very happy-seeming is.
嬉しそうだ is the visual-inference そう attached to the i-adjective stem 嬉し-. This is the 様態 (manner) そう — 'X-looking / appears X' — built by dropping the い from an i-adjective and attaching そう. Distinct from the hearsay そう, which attaches to the full plain form (嬉しいそうだ would mean 'I hear she's happy'). With visual そう you've seen something that suggests the state; with hearsay そう you've been told.