Why is K the equilibrium constant changed if there is a temperature change in the system?

Let's look at the effect of temperature changes from a Le Chatelier perspective. I'll use a simple exothermic reaction at equilibrum:

A ---> B + heat

Increase the temperature and the reaction responds by moving the position of equilibrium back to the left (increasing the conc of A and decreasing the conc of B).

Now, look at the consequences of the effect on the K.

K = [ B] / [A]

Adding the heat caused the value of the numerator to go down and the value of the denominator to go up. The net effect is to cause the value of the K to go down.

If I were to decrease the temperature, the opposite to the above would happen. (You may explore the consequences on an endothermic reaction on your own).

The reason for this is that there is not a term in the K equation for the heat, so an inc/dec in temp in the reaction is not compensated for in the equilibrium constant equation.