Q:

What is a rocket engine thrusting against in space?

I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity?

A:

Newton's third law is pretty near to the mark.
All of the phenomena you cite stem from the principle of conservation of momentum in an isolated system, itself ultimately a result (through Noether's theorem) of the fact the physical description of that isolated system is unchanged if we shift the spatial origin of our co-ordinate system.
So, if you're in deep space and you throw something with mass $m$ in one direction at a speed $v$, its momentum is $m\,v$ in that direction. The initial total momentum of the system (you + the thrown thing) is nought. So that means that the final total momentum for the system must be nought. Therefore, your own momentum must be $m\,v$ in the direction opposite to the thrown thing. If your mass is $M$, then your speed is $m\,v/M$ in the direction opposite to the thrown thing.
Note that, even though you can't shift your centre of mass without throwing anything (and in any case, the centre of mass of the whole system i.e. you+the thrown thing stays put), you can rotate and shift your orientation without violating conservation of angular momentum by cyclically shifting your shape; this is the same method a cat uses to flip over as it falls. See my answer here to the question "Is there a way for an astronaut to rotate?" and also my article "Of Cats and Their Most Wonderful Righting Reflex"
General relativity doesn't describe rockets (well, almost so, see my caveat below) in the way you might think. General relativity describes the locally freefalling frames and their so called Lie dragging by the system of geodesics defined by the solutions to the Einstein Field Equations. In less jargon: General Relativity tells you what kinds of motions are in keeping with Newton's first law; it tells you the motions within spacetime that something "isolated" (not experiencing a force) will undergo: anything different to this calls for a force to accelerate something relative to these freefalling frames. Chemistry describes the burning of fuel and Newton's third law the production of force from throwing this fuel to allow your rocket to deviate from the freefalling motion given by General Relativity.
To be precise, as the rocket throws its fuel out, the mass-energy distribution and the momentum fluxes (pressure distributions) of the system is changing, and this strictly speaking would need to be taken into account in the Einstein Field equations (this would alter the "source" term, the so-called stress-energy tensor). Thus the rocket's action would, to a fantastically small degree, alter the spacetime around it. But this is a tiny technicality. The main gig is simply that chemical energy allows you to throw fuel and produce a force to let you deviate from a locally freefalling (inertial) frame.

A:

I'm a little confused why you ask about GR at the very end of your question.  If your question is simply how a rocket is able to accelerate in space without having anything to "push" off, then we can tackle your question pretty well with classical mechanics.
Newton's Third Law has its limitations at times, but for this question it will work just fine.  Newton's Third Law states that for any force exerted by object $A$ onto object $B$, then object $B$ exerts a force back on object $A$ of equal magnitude, but in the opposite direction.  In this example, the rocket accelerates, so some object $A$ must be exerting a force forward on the rocket.  In turn, the rocket must exert a reaction force back on this mystery object $A$.
It turns out that object $A$ is the gases escaping the rocket.  As the gases escape the nozzle of the rocket, they interact with the rocket itself.  The gases exert a force forward on the rocket, and the rocket pushes back on the gases because of Newton's Third Law.  Although it may seem strange, this is all you need to accelerate your rocket.  Your rocket does not need ground, or an atmosphere, or another space vehicle to "push off"; the interaction between the gases and the rocket occurring within the rocket is sufficient to accelerate the rocket.
If you don't believe me, sit on a chair or a scooter on a low friction surface, at rest.  Hold a heavy object like a bowling ball or medicine ball on your lap. As hard as you can, throw the ball away from you.  If the coefficient of friction between your chair and the floor is low enough, you will experience an acceleration in a direction opposite of the ball.  Because of Newton's Third Law, the ball exerted a force back on you, and if the friction is low enough, there is a net force in that direction that produces an acceleration.
As a bonus, please note in both of these examples that momentum will be conserved. so no cause for panic there.

A:

If I'm not wrong, it's basically the same principle in which an astronaut would throw something in empty space and, with so, move in the opposite direction.
It's not thrusting against something but throwing energy and power by burning fuel according to the law of inertia... I could be wrong so I'd like someone more knowledgeable to double-check, please :)

