。
She is looking forward to her birthday party.
Literal
She [topic-は] her own birthday-party [object-を] enjoyment [as-に] is-doing.
楽しみにしている is the resultative state of 楽しみにする ('to make X into one's enjoyment' / 'to look forward to X'), built on the productive [X]を[Y]にする pattern. It's the standard, idiomatic Japanese way to say 'looking forward to something.' 誕生パーティー mixes the Sino-Japanese 誕生 ('birth') with the loanword パーティー — a hybrid common in modern event vocabulary; お誕生日会 is the older purely-native equivalent.