It’s a good question. You’d think that a rocket’s exhaust has to push against something, like a paddle needs to push against water to propel a canoe.
But in reality, a rocket doesn’t need to push against the atmosphere to fly. What it’s doing is essentially pushing against the inertia of the rocket exhaust.
But that’s confusing. Let’s try to look at it from a more earthbound angle.
Let’s say you’re sitting in a shopping cart with a bowling ball in your hands. You lift the bowling ball over your head and toss it as hard as you can toward the back of the cart.
If you’re on smooth pavement with well-oiled wheels, your cart is going to move forward at a good pace.
If someone hands you bowling ball after bowling ball for you to toss in quick succession, you’ll be able to keep your cart moving forward. You’ve created the equivalent of a slow-motion rocket engine.
But remember that the reason you’re moving is not because the bowling balls are pushing against the air, it’s your action of tossing them that’s pushing the cart forward.
That would be true even in the vacuum of space.
What happens with rocket engines is that as rockets burn fuel, they release a huge collection of gases out the back at a tremendous speed.
The effect in forward motion is the equivalent of chucking thousands of bowling balls a second.