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She started playing tennis more out of vanity than curiosity.
Literal
She [topic-は] curiosity from [than-というより] vanity from tennis [object-を] began.
~というより ('rather than, more like') is a comparative qualifier, used when the speaker wants to revise or refine an apparent label: 'X — or more accurately, Y.' The frame [A]からというより[B]から (literally 'from A as a source, more like from B as a source') sets the contrast cleanly: 好奇心から ('out of curiosity'), 見栄から ('out of vanity'). 見栄 ('vanity, showiness, putting on airs') is the social impulse to look good in others' eyes — 見栄を張る ('keep up appearances') is the standard verb collocation. The judgment edges toward gentle critique.