What exactly does "sweep me off my feet" mean? (And why?)

Solution 1:

Although the phrase can mean that, and often does, it's also sometimes applied in a more broad context. To be "swept off your feet" is to be surprised, enthralled, exhilarated. Critics can be swept off their feet by an epic film; operagoers can be swept off their feet by a beautiful aria, etc.

As for how sweeping became associated with love, that's referring to the aspect of sweeping that means a smooth movement, not the act of using a broom. Ballroom dancers can sweep across the dance floor, a powdery snow can sweep across the barren fields. It's that smooth, fluid motion – and the idea of your emotions being carried in that fashion – that brought about the idiom. A strong ocean or river current can literally sweep you off your feet, and young lovers can do the same thing to each other, figuratively and emotionally.

Solution 2:

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

So "Are you trying to sweep me off my feet?" translates to, literally, "Are you trying to make me fall (in love) with you?"

It's like making someone fall in love with you in a short amount of time.

Source Urban Dictionary