The answer has to do with entropy (the amount of disorder that a substance has). The melting point of a substance is the temperature at which the solid and liquid form of a substance are at equilibrium. From thermodynamics, we know that equilibrium corresponds to a state where ΔG = 0.
Since ΔG = ΔH - TΔS, setting ΔG = 0 gives us the relation:
Teq = ΔH/ΔS
So the melting point depends on the ΔH and ΔS of a process.
What happens when we add salt to pure water? Adding salt to water increases its entropy, because it is now a mixture instead of a pure substance. Since ice already has a lower entropy than water (because ice is ordered into a crystal lattice and water is not), increasing the entropy of water makes the difference between the entropy of ice and water greater (i.e. it increases ΔS). According to the relation above, increasing ΔS should decrease Teq.