She's beside herself with joy.

Literal

She [topic-は] joy [from-で] self [object-を] is-forgetting.

我を忘れる ('forget oneself') is a vivid idiom for being lost in something — a state where consciousness of self drops out, eclipsed by the absorbing emotion or activity. The 我 here is an old-fashioned, slightly literary first-person pronoun ('self, I'); modern speech rarely uses it standalone but it survives in idioms (我が国 'our country,' 我ら 'we'). The casual ~てる contraction of ~ている keeps the register relaxed and present-tense, painting her as currently in that out-of-self state.