。
She began talking to the dog.
Literal
She [topic-は] that dog [to-に] began-to-talk.
~始める is a workhorse compound suffix — attach it to the masu-stem of any verb and you have 'begin to X': 話す → 話し始める, 食べる → 食べ始める, 読む → 読み始める. The に here marks the addressee — 'talk *to* the dog' — distinct from と, which would mean 'talk *with* the dog' (a two-way exchange). The dog presumably isn't holding up its end of the conversation, so に is exactly right.