。
She is enormously proud of her daughter.
Literal
She [topic-は] her own daughter [object-を] greatly is-boasting-about.
大いに ('greatly, considerably, much') is a mid-register adverb — slightly more written than とても, but less stiff than 非常に. 自慢する takes its object with を and pairs naturally with ~ている for an ongoing attitude. The reflexive 自分の specifies it's her own daughter rather than someone else's — without it, the sentence could be misread as her boasting about a daughter in general.