Here's why:
* Protons in the nucleus have positive charges and repel each other. This repulsion should cause the nucleus to fly apart.
* The strong nuclear force is much stronger than the electromagnetic force (which causes repulsion). It acts over very short distances, only within the nucleus, and overcomes the electrostatic repulsion between protons.
* This force also binds neutrons to protons and neutrons to each other. It is the strongest force in the universe, and is responsible for the stability of atomic nuclei.
Let me know if you'd like to know more about the strong nuclear force!