Crudely, just consider the burned rocket fuel that is ejected out of the back of the rocket due to chemical reactions. The rocket exerts a strong backward force on the burned rocket fuel. According to Newton's third law, required reaction is that the burned rocket fuel exerts an equal forward force on the rocket. This force accelerates the rocket forward.
Keep in mind that Newton's third law says nothing about pushing against something, the rocket does not need to push against "a medium" to accelerate forward.
Why do you want to know?
I'm not kidding. That's actually an important question. The answer really depends on what you intend to do with the information you are given.
Newton's laws are an empirical model. Newton ran a bunch of studies on how things moved, and found a small set of rules which could be used to predict what would happen to, say, a baseball flying through the air. The laws "work" because they are effective at predicting the universe.
When science justifies a statement such as "the rocket will go up," it does so using things that we assume are true. Newton's laws have a tremendous track record working for other objects, so it is highly likely they will work for this rocket as well.
As it turns out, Newton's laws aren't actually fundamental laws of the universe. When you learn about Relativity and Quantum Mechanics (QM), you will find that when you push nature to the extremes, Newton's laws aren't quite right. However, they are an extraordinarily good approximation of what really happens. So good that we often don't even take the time to justify using them unless we enter really strange environments (like the sub-atomic world where QM dominates).
Science is always built on top of the assumptions that we make, and it is always busily challenging those assumptions. If you had the mathematical background, I could demonstrate how Newton's Third Law can be explained as an approximation of QM as the size of the object gets large. However, in the end, you'd end up with a pile of mathematics and a burning question: "why does QMs work." All you do there is replace one question with another.
So where does that leave you? It depends on what you really want to know in the first place. One approach would simply be to accept that scientists say that Newton's Third Law works, because it's been tested. Another approach would be to learn a whole lot of extra math to learn why it works from a QM perspective. That just kicks the can down the road a bit until you can really tackle questions about QM.
The third option would be to go test it yourself. Science is built on scientists who didn't take the establishment's word at face value, went out, and proved it to themselves, right or wrong. Design your own experiment which shows Newton's Third Law works. Then go out there and try to come up with reasons it might not work. Test them. Most of the time, you'll find that the law holds up perfectly. When it doesn't hold up, come back here with your experiment, and we can help you learn how to explain the results you saw.
That's science. Science isn't about a classroom full of equations and homework assignments. It's about scientists questioning everything about their world, and then systematically testing it using the scientific method!
Best Answer
Newton's third law states that if object A acts on object B with force $\mathbf{F}_{AB}$, then object B must act on object A with force:
$$\mathbf{F}_{BA}=-\mathbf{F}_{AB}$$
When expressed in terms of A and B's momentum, the same equation can be written as:
$$\frac{\mathrm{d} \mathbf{p}_A}{\mathrm{d}t} = -\frac{\mathrm{d} \mathbf{p}_B}{\mathrm{d} t}$$
Rearranging:
$$\frac{\mathrm{d}}{\mathrm{d} t} \left [ \mathbf{p}_A + \mathbf{p}_B \right ] = 0$$
Or:
$$\mathbf{p}_A + \mathbf{p}_B = constant$$
This says that A and B must act on each other such that the sum of their momenta is constant over time. So Newton's third law is just an expression of the conservation of momentum. Momentum conservation itself can be seen as a consequence of spacial translation symmetry, as shown by Noether's Theorem. This symmetry remains in quantum mechanics, so momentum conservation still applies in QM.