At its most basic level, Newton's Third Law is just a statement of the conservation of momentum. Force is a measurement of change in momentum per unit time. If two particles interact, the change in momentum of particle 1 has to be equal and opposite to the change in momentum of particle 2. This means that the interaction forces have to be equal and opposite.

If you want to go deeper, the fundamental forces are all generated by particles exchanging "virtual" particles. These virtual particles carry momentum, and so the real particles can push/pull on each other by the exchange of virtual particles. Since momentum is conserved, the momentum gained by one real particle has to be equal and opposite to the momentum gained by the other real particle. If you think of these momentum changes as forces, this is Newton's third law.