The answer lies on strong nuclear force.
This type of force holds protons and neutrons together. Protons do exert this force with each other but electrostatic repulsion is stronger. Neutrons also exert this force but do not repel each other. Adding neutrons to the protons help them to overcome the repulsion, allowing them to stick together. Essentially, neutrons stabilize the nucleus ONLY at short distance (this force is only stronger than electrostatic repulsion at short distance) by attracting each other and protons equally with strong nuclear force.
However, as protons increase, more neutrons are needed to have strong nuclear force to overcome the repulsion. Eventually, atoms reach a point in which the repulsive force becomes way too huge for strong nuclear force to handle. This causes the nucleus to break apart into smaller nuclei, releasing a huge amount of energy in a process called fission. As a result, when you go down to the periodic table, you will notice that there are more neutrons than protons.
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