This phrase often signifies that emotions have overtaken logic or usual behavior. Sorry for the dramatic speech; I got carried away with the moment.
Pro Tip 2/3
Possible Regret
Sometimes suggests that one might later regret their actions done in excitement or passion. I got carried away buying gifts and now my wallet is empty.
Pro Tip 3/3
Colloquial Usage
Common in everyday conversation, but less appropriate in very formal or written contexts. He got carried away with the story and missed the bus stop.