So, if hypothetically two fermions were in the same state—for example, in the same atom in the same orbital with the same spin—then interchanging them would change nothing and the total wave function would be unchanged. However, the only way a total wave function can both change sign (required for fermions), and also remain unchanged is that such a function must be zero everywhere, which means such a state cannot exist. This reasoning does not apply to bosons because the sign does not change.