I'll just answer your original question. "Big time" is used to emphasise (or exaggerate) the phrase. You could also read it as "in a major way" or "really very much". It's a bit colloquial and slangy, so pay attention to when you can use it. "Big time" works as an adverb of manner, so no "a". Other examples: I won the lotto big time. She lost her temper big time. He's in trouble again, big time.

very high and important level of success in any given field

just means i owe you a lot , i owe you a big one。

I owe you big time mean I am really grateful, I owe you a lot.