Question

Why do atoms have an overall neutral charge?

Answer 1

This is because, Atoms generally or always have a neutral charge.
In the atom, I hope you're aware, lies the basic atoms Protons, Neutrons and Electrons. These you should know from GCSE.
Obviously there are more sub-atomic particles, such as, Neutrinos, leptons, baryons, etc.
The Protons 'match' the Electrons. Thus, meaning, the charge will be balanced; e.g. 4 Protons + 4 Electrons = +4 Positive charge + -4 Negative charge = 0 overall charge --> Neutral.
This is true unless, the atom is an Ion.
Hope this helped :)

Answer 2

Roughly speaking, for the same reason almost every adult man has a wife or girlfriend at any given instant of time: because if he does NOT, he exerts all his force to find one, and only stops searching when he does.

It's the same with a charged particle, like an atom missing an electron. It exerts such a powerful electric force on any oppositely charged particle nearby, that it very rapidly finds a partner, and this powerful electric force only stops when it's "satisfied" by their being exactly as much negative charge as positive charge in the atom.

You might wonder how it is that there actually is always a negative partner for every positive partner -- why no atoms have to be "bachelors" forever. We don't actually know why, it's just a experimental fact about our universe, that there appears to be exactly as many positive particles as there are negative particles.