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1.
It's an english expression reffering to the feeling that one gets when completely taken by someone, carried away, swept away (all emotionally).

It's basically like making someone fall in love with you in a short amount of time.
so "are you trying to sweep me off my feet?"
basically translates to, literally
"are you trying to make me fall (in love) with you?"
by dagirlshommie April 26, 2010