I know that according to dictionaries and the answers here the meaning of this phrase ranges from the sincere Pardon me, or I am sorry for what I just did to the more sarcastic or sometimes even aggressive I beg your pardon?.
Yet I cannot find the words to describe the phrase’s meaning (nuances) in Lynn Anderson’s song, “Rose Garden”.
Best Answer
In order to understand I beg your pardon, especially but not exclusively in the context of the song, it can help to look at it literally:
Beg is stronger than ask, but pardon is usually a lot less so than forgiveness (unless it's the kind of pardon that gets you out of jail or reprieved from a death penalty...).
It fits in the sense of I am sorry for what I just did, because you ask for forgiveness for your recent action.
In the song, it fits, because in the song, it is also a veiled apology:
Obviously, the rose garden is a common symbol for an easy and beautiful life (or relationship).
In a more matter-of-fact way, the lyrics could be rephrased as: