。
She was admitted to the club.
Literal
She [topic-は] that club [in-に] joining [object-を] was-permitted.
許される is the passive of 許す ('to permit, allow, forgive'). The に here marks the entity being joined: ~に入会する ('to join, enter membership'). 入会 is itself a Sino-Japanese compound — 入 ('enter') + 会 ('association, group') — and parallels 入学 ('school enrollment'), 入社 ('joining a company'). The を on 入会 marks her membership-entry as the thing she was permitted to do; this kind of nominal-as-object passive ('the act of joining was permitted') is more formal-sounding than the simpler 入会できた ('was able to join').