tldr:
It's a matter of switching your viewpoint; that is, it's a matter of changing what we view as the "system" and calculating appropriately. In addition, thinking about this as the "forces canceling out" is not really the best way to think about things. Instead, you should be thinking about the outcomes of those forces acting by determining whether or not the the impulses or internal works done on the objects inside the system by each other cancel.
When we think of two objects 1 and 2 exerting forces on each other, and we care only about the motion of one of the objects (object 1, say), then we consider only the forces acting on that object. To figure out what happens to object 1, we can compute the impulse imparted to object 1 by object 2 as
$$
\vec{J}_{\textrm{2 on 1}}=\vec{F}_{\textrm{2 on 1}}\Delta t\,,
$$
where $\Delta t$ is the amount of time that the force acts (I am assuming for simplicity that the force is constant; we would need an integral otherwise).
In the case where this is the only force acting on object 1, this is equal to its change in momentum,
$$
\Delta \vec{p}_1 = \vec{J}_{\textrm{net on 1}} = \vec{J}_{\textrm{2 on 1}}\,.
$$
Similarly,
$$
\Delta \vec{p}_2 = \vec{J}_{\textrm{net on 2}} = \vec{J}_{\textrm{1 on 2}} = \vec{F}_{\textrm{1 on 2}}\Delta t\,,
$$
again assuming that the only force acting on object 2 is $\vec{F}_{\textrm{1 on 2}}$. That's the end of the story, as far as it goes. We can now compute what happens to the two objects individually as a result of their collision with each other.
However, it is often the case that we know very little about the details (which is to say, the forces) of the collision between the two objects. If we can find a way to eliminate these forces from the calculation, then we don't need to know these details, and that can help us. So we note the following interesting phenomenon, which is a consequence of the third law:
$$
\Delta \vec{p}_1 + \Delta \vec{p}_2 =
\vec{F}_{\textrm{2 on 1}}\Delta t + \vec{F}_{\textrm{1 on 2}}\Delta t
= \vec{F}_{\textrm{2 on 1}}\Delta t +(-\vec{F}_{\textrm{2 on 1}})\Delta t
=0\,.
$$
Again assuming that no other forces act during the collision (or, more true-to-life, they're small enough that we can neglect them), the sum of the changes in momenta of the objects is zero!
For this reason, we redefine our system to be composed of both objects together, and we define the total momentum $\vec{P}$ to be $\vec{p}_1+\vec{p}_2$. Our calculations show that even though the objects are exerting forces on each other during the collision$-$and thereby changing each others' momentum$-$the total momentum of the system is left unchanged. This is what is really meant by "the internal-to-the-system forces cancel each other out during the collision".
So, just to comment, then: the point is that we need to be careful and specific about what we mean when we say the forces "cancel each other out", because they "can" in one context and not in another, but that's because we are computing different things. (Really, we shouldn't talk about the forces canceling each other out; instead we should be talking about the impulses canceling, or, in an energy context, the internal works canceling (or not).)
To add fruit to the fire, to coin a mixed metaphor, things are different again when we are thinking about the work done during collisions, i.e., when we are taking an energy perspective on the system rather than a momentum perspective.
If two objects are colliding and exerting contact forces on each other (like two billiard balls), then one of the objects does positive work and the other does negative work, and these works are equal and opposite, so there is no change in kinetic energy (assuming the collision is elastic).
However, if two objects inside the system are interacting via conservative long-range forces like the gravitational force, then the two objects can both do positive work on each other, increasing the kinetic energy of the system! So in this latter case, the forces don't "cancel out", even though they are equal and opposite forces acting on two parts of a single system. But, we still need to be careful! The forces do "cancel out" in the sense that the total momentum is still conserved, because the changes in momentum are equal and opposite.
