He had no intention of letting her drive the car, but she begged so persistently that he finally gave in.
Literal
He [topic-は] her [to-に] car [object-を] drive-[causative-させる] intention [topic-は] not-was [explanatory-のだが], she [subject-が] too-much begged [reason-ものだから], he [possessive-の] side [subject-が] finally yielded.
運転させるつもりはなかった 'had no intention of making her drive' — causative + つもり. あまりにせがむものだから 'because she nagged so much' — ものだから is a emphatic causal pattern. とうとう折れた 'finally gave in/yielded' — 折れる literally 'to break/bend.' A relatable domestic negotiation.