When Finn arrives on Starkiller Base, he’s wearing Poe’s leather jacket. Later, when he and Rey run to open the doors for Han and Chewie, Rey is wearing it. A few scenes later, watching the Han and Kylo showdown happen, Finn puts the jacket back on.
The switch occurs during the deleted "snow speeder chase" scene. The only info we get in any of the script, the novelisation or the junior novelisation is this very brief exchange:
“Snow is cold!” Rey squeezed the speeder between a phalanx of willowy alien trees. “It’s the complete opposite of Jakku!”
“Try living here,” Finn told her. “There are only two seasons: winter, and dead of winter!”
I think we can assume that he lent her his jacket because she was cold and unused to the extremely low temperatures on the planet. You can see him wearing the jacket in the top shot (before they mount the speeder), then shortly afterwards (in the lower shots), she's wearing the jacket.
She was cold, and he was being a gentleman in letting her borrow his jacket. Then he got it back. Simple as that.