ELI5: How does blowing a large flame out of the bottom of a rocket make it fly? Why Fire instead of just air?

The shear amount and speed of exhaust gasses push the rocket up.

It’s not blowing a flame, like the burner on a gas stove.

It’s a giant continuous explosion, like the kind that propels a bullet from a gun.

It's not the fire that is making it fly, it's the energy - we just see some of it in the visible spectrum.

The whole idea of a rocket is to throw stuff out the back as fast as possible. Burning fuel is good at that because it adds a lot of energy, but you absolutely can use "just air". That's called a cold gas thruster. They don't perform well, definitely not well enough to get into space but they're really easy to use, and some space ships use them to point themselves in the right direction.

Same reason you can ball up some paper and throw it halfway across the room with a straw. The fire is just a byproduct of the explosive expansion of gas that's pushing off the rocket against the ground.

An explosion is a large, quick release of energy resulting in expansion of gasses. A rocket is basically riding a continuous explosion.

The gas particles need to move extremely fast to have enough momentum to move the rocket. The explosive expansion of rocket fuel with an efficiently shaped nozzle provides that speed. How would they accelerate the air out of the nozzle?

Hot air expands and when it expands in a confirmed space it creates pressure. Create a hole to exit that space and you have propulsion

In order for the rocket to go up fast something from the rocket needs to go down fast. You can use pressurized air but thats not enough stuff going fast enough for long enough. Burning fuel lets enough stuff go fast enough for long enough that the rocket reaches space.

They do only blow gases, and that is what makes the rocket move. The fire is how the rocket generates such a huge amount of gas fast enough to fly.

Liquids are denser than gas, so when they burn they increase in size.

This pushes gas out the exhaust.

Action equals reaction. Gas flowing back means rocket flowing forward.

Everything has an equal and opposite reaction. The flames are pushed backward so the rocket is pushed forward.
As for why fire, the heat from fire makes the gasses expand faster and harder, thus pushing the rocket faster and harder.