You're studying with Headway Upper Intermediate, right? "the second time around for both of us" in this context means each of them were married before. You can add "it being..." to the end of a sentence. The meaning is "because it's..." Pack a sweater for your trip to Australia, it being winter there this time of year.

It's a phrase, not a sentence. It should be written, "That's fine. He's invited too, of course. Ben and I want a really quiet affair, IT BEING the SECOND TIME AROUND for BOTH OF US". The "second time around" means they had each been married previously.