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This question already has an answer here:

I don't know maybe it's a very stupid question I'm asking but that's the question which always disturbed me.

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marked as duplicate by John Rennie, AccidentalFourierTransform, David Hammen, JMac, dmckee Feb 7 '18 at 20:17

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Rockets push on the exhaust that they make when rocket fuel is burned. It's exactly the same process that creates recoil in a gun; in fact, there's a pretty well-known informal analysis that examines just how many machine guns you would have to fire at the ground to make a rocket (well, in the analysis they compare it to a jetpack, but the principle is the same: the jetpack pushes on its exhaust): https://what-if.xkcd.com/21/

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Think of it in terms of conservation of momentum. When propellant is expelled in the direction opposite of travel, the rocket has to make up for this "negative" momentum and gain velocity in the positive direction.

You also have to take into account the change in mass of your rocket. You're equation becomes:

$$\frac{d(mv)}{dt}= F + v'\frac{dm}{dt}$$

Where $v'$ refers to the velocity of the ejected mass.

Reference:

https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec14.pdf

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