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Equal and opposite reaction.

There's a law for this. The matter is "pushing" against the ship, it doesn't have to push against anything else.

In fact having an atmosphere to push against actually reduces the effectiveness of thrust due to atmospheric pressure, which must be overcome. Which is why different engines are designed to run in atmosphere versus out of atmosphere.

If you throw a baseball in space you have transferred momentum to that baseball, pushing you back. You will move in the opposite direction (likely spin because you just imparted angular momentum onto yourself since you didn't throw from center of mass)

A rocket does not work because there is "something to push against". Your initial assumption is flawed, so it's normal and expected that you cannot reason about this right now. A rocket works because there are gases that are ignited, looking to expand. This expanding force is applied to the nozzle, hence to the entire body of the rocket, and pushes it in the opposite direction : up

Imagine yourself floating in space with a heavy object held in your hands. Say an anvil. You push the anvil away. This gesture is going to push you back by some amount as well, since the anvil is so heavy. Well the rocket is you, and the burning fuel is the anvil. A rocket is just an object continually jettisoning weight behind it so it can move forward

Every action has an equal and opposite reaction

Think of it like this, imagine you are in space and you throw a baseball in front of you, this action will cause an opposite reaction, moving you backwards

A lot of people are offering explanations, but I think I'm going to give one too.

Think of recoil in a gun. If you don't have a mental image of it, watch a few youtube videos of people firing handguns. Look for videos of big, high-recoil handguns, like the Desert Eagle or the Magnum (or the Super Ruger Redhawk according to chat-GPT).

You need to get a good look at handguns pushed backwards as they are fired.

Now think about this: those bullets aren't pushing against an atmosphere. They are pushing only against the inside of a gun.

But when this tiny, tiny bullet pushes super-fast against the gun, using the gun to accelerate to incredibly high speeds very quickly... it pushes the gun really hard in the other direction.

Get that mental image into your head. Small object can push large object with a lot of force by kicking off of large object with insane speed.

Now: Take away the person holding the gun. Take away the planet. Take away the atmosphere. Put that gun in space and pull the trigger again. (Just make sure to use a gun that has modern ammunition that doesn't require oxygen to fire).

What happens to all that recoil? What does the recoil do to the gun now? The bullet still goes flying out of the chamber. Still does this by pushing against the gun.

Hopefully it should now be easy to imagine that the gun will start moving.

Rocket fuel is basically a tank full of bullets.

The main function of rocket fuel is "heavy stuff that is shoved out of the spaceship to make it move."

The reason we use highly explosive fuel is because "shoving heavy stuff away from you at the speed of a bullet" is going to move you more than "shoving heavy stuff away from you at normal speed."

Does this make any sense?