Why do protons contain a positive charge? Why do electrons contain a negative charge?
I'm not sure there is a meaningful answer to this question, at least not one that is known to current science.
We can explain why we call them positive and negative. It was basically an arbitrary decision, and we probably should have gone the other way around because electrons are a lot more mobile. The "positive" terminal of a battery is the one that's missing electrons. This is a very odd convention.
We can even sort of explain why they are opposite. There's a fundamental balance between electric charges in the universe called the CPT (charge-parity-time) symmetry. So they have to be exactly equal and opposite.
In the particular case of the proton, we can break it up into quarks (well, theoretically; we can't actually separate them for more than a few nanoseconds) and say that a proton is two up quarks, each with +2/3 charge, and one down quark with -1/3 charge, and therefore a total of +1 charge. (No such luck with the electron; leptons have no constituent parts so far as we know.) Even then, we still don't know why up and down quarks have those charges.
But why is there electric charge at all? Who do we have these four forces, and not some different set of forces? Why does matter exist, and why hasn't it been annihilated by an equal quantity of antimatter? Why is the mass of the proton so much smaller than its charge (this question makes sense if you use natural Planck units)?
Nobody knows. This is the frontier of modern theoretical physics. String theorists think they have an answer, but so far they haven't come up with a theory we can actually test in experiments.
Come back in 100 years, and maybe we'll have it figured out by then.