。
She entered her terrier in a dog show.
Literal
She [topic-は] her own terrier-dog [object-を] dog-show [to-に] put-out.
出す ('to put out, send out, submit') in the sense of 'enter (a competition), submit (an entry)' is a common collocation: 大会に出す ('enter into a tournament'), 申請を出す ('submit an application'). The に marks the venue or destination. テリア ('terrier') is a loanword; 犬 + 品評会 = 'dog show, dog exhibition.' 品評会 ('judging/exhibition event') applies to plants, animals, and crafts being formally compared at organized shows.