。
She began to give up on success.
Literal
She [topic-は] success [object-を] give-up began.
あきらめ始めた uses the ~始める (はじめる) suffix meaning 'to begin to' — attached to the masu-stem of あきらめる (to give up). The compound marks the onset of the process: she started giving up, not that she fully gave up yet. 成功 (せいこう) means 'success.'