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She had to make her son withdraw from the club.
Literal
She [topic-は] son [object-を] that club [from-から] make-withdraw had-to.
Two layered grammar points. 脱退させる is the causative of 脱退する ('to withdraw, leave a group'), so 'make/cause [him] to withdraw.' ~なければならない is the standard 'must / have to' obligation pattern; in past tense ~なければならなかった ('had to'). The combined structure expresses a forced, reluctant decision — 'she had to compel her son to leave.' The choice of 脱退する (rather than 抜ける) tilts the register slightly formal; 脱退 implies more deliberate withdrawal from an organization.