。
She takes great pride in her stamp collection.
Literal
She [topic-は] her own stamps' collection [object-を] very-much pride [as-に] is-doing.
切手 = ki-tte ('postage stamp'); 切手の収集 ('stamp collection / collecting of stamps'). 自慢にする here is the productive [X]を[Y]にする pattern — 'treat X as one's pride.' 大変 ('very, greatly') intensifies the attitude. Stamp collecting (philately) is a long-running hobby internationally; in Japanese it's also expressed as 切手収集 or just 切手集め in casual speech.