The above sentence refers to a situation where you are in a crowded subway train. You know, when you are in a crowded subway, it feels like you stand up being supported by people around you who pack into the subway train instead of standing up with your feet on the floor.
떠 있다 means "to stay up in the air". He/she is describing the situation where he/she are in a crowded subway train which so many people pack into. He/she feels like stayin up in the air.
~듯한 기분이 들다 means "feel like ~~" This is very frequently used expression.
for example, 세상을 다 가진 듯한 기분이 들었다.(I feel like I own all the world, this is often said by a groom in a wedding ceremony when he finally marry his girlfriend)
I hope that I have you understood. Sorry. I am not good at Englishㅠㅠ