When a ball is thrown onto a wall, the ball exerts a force onto the wall. According to Newton's Third Law, the wall will exert an equal and opposite force to the ball. Thus, how would the ball be able to bounce back? Shouldn't it just slide down the wall?
You have to consider this as a dynamic process. To make sense of it you also have to think of the ball as NOT a point particle, so it has a centre of gravity, probably in its actual centre. As the surface of the ball impacts the wall the centre of gravity slows down, which means its momentum drops. (It also means that the wall or the ball or both must deform during the impact) Newton's second law requires that momentum can only change if there is an external force applied and this comes from the wall. This force increases from zero as the impact proceeds, it is exerted in the opposite direction of the balls motion. At some point the momentum will reach zero, but the force is still there and still directed outwards so now it begins to accelerate the ball's centre of gravity back in the opposite direction. As it moves out the balls momentum increases and the force drops but still points in the same direction. So the ball accelerates until the surfaces separate at which point the force is back to zero.
Now it is true that Newton3 requires that throughout this process the ball is exerting an equal and opposite force on the wall but the point here is that these two forces are exerted on DIFFERENT objects. So the force of the ball on the wall acts to try and increase the momentum of the wall (if it was somehow moveable it would accelerate away from the impact). For a fixed wall the force goes to make some deformation of the wall (bending it / making a dent in it etc.)
If both objects are perfectly elastic, meaning they return to their original shape once forces are removed, then the ball will rebound with exactly the same but opposite momentum it had before the impact. If either object is permanently deformed in the impact, the ball will rebound with less momentum.
The forces don't anulate each other because they are not on the same body. The ball creates a force F on the wall. Then, according to the Newton's Third Law, the wall will create a force N with the same module of F but with oposite direction. Pay attention that F is on the wall and N is on the ball, that's why you can't sum them.
(The other answers have focused on the fact that the equal and opposite forces of Newton's law are not acting on the same object, rather the ball on the wall and the wall on the ball. However, as I see it, the question already takes that for granted, but wants to know why that opposite force is enough to rebound the ball and not just stop its motion)
The question assumes that because the ball is moving with a momentum of $mv$ (mass-of-ball x velocity), the force it applies to the wall is also $mv$. However, this is incorrect. The actual force given is $2mv$, so the 'equal and opposite force' on the ball will be $-2mv$ - just enough to stop the ball and send it back at the same speed.
Keep in mind that when we say $2mv$ is applied to the wall, the $m$ is the mass of the ball, so relative to the mass of the wall (and the earth it's attached to) the force $2mv$ is tiny and will only move the wall/earth an immeasurable amount.
To be really accurate, the force is not exactly $2mv$/$-2mv$, because that would be a violation of conservation of energy - if the ball rebounds at the same speed, how could any energy be transferred to the wall? Instead, it's a bit less than $2mv$; The exact value has to be calculated to obey both the rule of conservation of momentum and conservation of energy. (This point has been discussed further here).
The fact the the force given is not the same as the momentum of the ball may be surprising, but because energy and momentum are conserved no laws of physics are violated. [Here's a hint: from the perspective (or: rest frame) of the ball, the wall is rushing towards it with a tremendous momentum of $2Mv$ ($M$ being the mass of the wall/earth) so it would expect to fly off at tremendous speed, upon collision. Obviously the force does not necessarily correspond to the momentum.]