Although the phrase "sweep me off my feet" probably means, "make me fall in love with you in a short time", what does it exactly mean, because "sweeping" can be difficult to be associated with "love". (It can be difficult to read the words "sweeping" and "feet" to get a feeling that it means love).
Below is one of its usage in Steve Jobs's letter to his wife:
We didn’t know much about each other twenty years ago. We were guided
by our intuition; you swept me off my feet. It was snowing when we got
married at the Ahwahnee. Years passed, kids came, good times, hard
times, but never bad times. Our love and respect has endured and
grown. We’ve been through so much together and here we are right back
where we started 20 years ago—older, wiser— with wrinkles on our faces
and hearts. We now know many of life’s joys, sufferings, secrets and
wonders and we’re still here together. My feet have never returned to
the ground. — Steve Jobs
Best Answer
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.