There really isn't a good answer, besides "that's how it is".
Each personal pronoun has two object forms: stressed and unstressed:
her. - Eu o înțeleg pe ea.
However, the stressed form is optional, the unstressed one is mandatory and it's also needed with nouns:
I understand her. - Eu o înțeleg.
I understand the girl. - Eu o înțeleg pe fată.
The "pe" is even weirder. It has no lexical meaning, you could think of it as a helper needed by words when used as objects, like in the sentence at hand. Section 126.96.36.199 of this pdf attempts to clarify when to use it.
The voice here in the dicsussion is clear but there is a new voice about which makes this sound like "eu me înțeleg pe fată" I hope DL sorts this cvoice out soon as there are lots of examples like this
Could someone explain why the indefinite form 'fată' is used here, instead of the definite form 'fata'?
When pe and other similar words like la or în are used the sort of contain the definite article themselves so the noun does not need to take the definite article ending, and never does. The major exception is the word cu which does not include a definite article and the noun the either does or doesn't take the article as an ending in order to distinguish, for example being with a girl or being with the girl.