Often you'll see force described as F=ma, but that's not entirely correct. Force is actually the time derivative of momentum, where momentum is p=mv, (mass times velocity), and using the product rule, F=mv'+vm'. v' is acceleration, and m' is change in mass. In most cases, we assume mass is constant, so the second term goes to zero, leaving us with F=ma=mv'.

In the case of rockets, the force on the rocket is equal to the thrust, and the sum of the forces is zero (it's in space). So plugging in our force equation, F=T=mu'+vm'=0, where m is the mass of the rocket, u is the velocity of the rocket, v is the exhaust velocity, and m' is the change in mass of the rocket. Rearranging this, u'=-v(m'/m), which tells us that rockets generate thrust because they expel their exhaust at high velocity, and they burn fuel at a tremendous rate (the change in mass).

So taking our last equation and multiplying by dt to get the differentials gives us du=v(dm/m), which after integrating gives us (u_2-u_1)=vln(m). If you think of it like compound interest, think of the mass of the rocket as the amount in the bank. You burn a bit of fuel to move the rocket, but this decreases the mass (changes the amount in the bank). You then burn a bit more fuel, which moves your rocket again, but because it's lighter than it was previously (there's less in the bank), the effect is different.