Why do two equal and opposite charges cancel out each other and become neutral? I have asked this from my teacher, and she told me that the charges cancel out because they are additive in nature. I want a better explanation. What makes them cancel their effect?
Assume that there is a $+q$ charge and $-q$ charge that combine and become neutral. When a third charge is brought close to the paired charge, it does not feel any electrostatics force by the paired charges, and we say that the paired charge is neutral. How do they become neutral? Doesn't the third charge feel the net electrostatics force by both +q and -q. The net force will only be zero on the third charge when the distance between the +q and -q with the third charge is the same, but its not possible every time. What makes them cancel out?
 A: You should be more precise as for what you call "effect". I'll assume that you mean electromagnetic field?
In this case, they don't cancel out. Charge is additive, yes. But your example of three particles is in fact very relevant, let's develop it.
First, let's take two opposite charges very close to each other. They form a neutral system called an electric dipole. The description of the electric field it generates is complicated, but it can usually be simplified with the following approximations:

*

*A rough approximation is to consider that, since they're opposite charges and very close, they're almost like a single neutral particle that doesn't generate any field. This is the monopolar approximation, not a very good one, but not stupid either.

*The most common approximation is the dipolar approximation, that I'll detail below. It's better than the monopolar one and is the one usually taught and used.

In this approximation, calculations are simplified by saying that we study only the field generated far from the dipole. It's a weak field because the system is neutral, but it's not zero because any point will be at different distances from both charges, so their respective electric fields don't cancel out.
This means that the dipole, even if neutral, has an electromagnetic effect on its environment, it can apply a force to a third charge particle that passes by (which will apply a force on it in return, of course).
A well-known example is the water molecule. While neutral, it acts as a dipole, which explain a lot of its chemical properties (the electromagnetic field it generates can influence other molecules, controlling how chemical reactions happen).
A: You are absolutely right. The net force will only be zero if the separation between the two charges is zero. The third charge will indeed feel the net force of the $+q$ and $-q$ charge and this will be slightly different if they are separated. This is what we call the higher moments of the force (you may have heard of dipoles as an example).
But your teacher alludes most certainly to the situation where the third charge is far away from the two original charges, so that that distance is much larger than the separation between these two charges. In that case the higher moments are very small and for all intents and purposes the third charge does not feel a net force; it sees only a neutral particle.  So your teacher is right as well.
You have discovered an important rule in physics: any statement one makes is only true in a certain context. Here the context is that the third charge should be far way from the two original charges.But it is in general always important to understand the context, i.e. the region of validity, of a statement one makes in physics.
