This usage of [optional] adverbial round is covered by oxforddictionaries...
4 So as to reach a new place or position, typically by moving to the other side of something:
He made his way round to the back of the building
They went the long way round by the main road
4.2 INFORMAL Used to convey the idea of visiting someone else:
Why don’t you come round to my flat?
The implications of including round in OP's example include the fact that although the speaker doesn't live permanently at his girlfriend's flat, it's not uncommon for him to be there (you wouldn't normally say you'd "I was round [somewhere] last night" if that was the only time you'd ever been there).
It's also unlikely you'd include round for some place that was a long way away (probably influenced by round/around = nearby).
There might be some debate as to whether OP's example is prepositional or adverbial. I don't see that distinction as useful here, but it may help to note that at, round and round at are all grammatically valid.
Essentially, you're round [at] some place round, around, near, nearby wherever you normally live (often, some place you regularly spend time at).