How does Koro-sensei fly?
How Koro-sensei can fly has never been mentioned in the manga or the anime, so I can only explain it based on my inferences from the manga.

Koro-sensei has been referred to as an octopus on multiple occasions in the anime and the manga (eg episode 10 at around 10:40 in the anime).
So it is fair to assume that Koro-sensei's body behaves like an octopus.

When you read about the locomotive system in octopi and the like, it's a jet propulsion system that they use to move around in water. Koro-sensei should be able to use similar methods in the air. As all his cells are tentacles1 and he can freely control his tentacles2, Koro-sensei should be able to fly with a jet propulsion system like the octopus.



1 Chapter 140, page 14.
2 Chapter 137, page 17.