Why does changing temperature change K?

K =	k_forward
	k_reverse

When a reaction is at equilibrium, the rates of the forward and reverse reactions are equal. The equilibrium constant is equal to the ratio of their rate constants (k).

Rate constants increase with temperature. For a particular equilibrium composition

increasing the temperature speeds up both the forward and reverse reactions.
decreasing the temperature slows down both the forward and the reverse reactions.

The equilibrium constant changes because for a particular reaction, the extent to which k_forward is changed differs from the extent to which k_reverseis changed.

Why is the extent of change of the forward and reverse rate constants with temperature NOT equal?
Equilibria are endothermic in one direction and exothermic in the other.
As shown in the diagram, the endothermic direction has a higher activation energy.
Rate constants for reactions with higher activation energies are more sensitive to temperature change.

Increasing the temperature causes in a bigger increase in the rate constant for the endothermic direction (and larger K)
Decreasing the temperature causes a bigger decrease in the rate constant for the endothermic direction (and smaller K)

Catalysts also change the rate constants.
Catalysts lower the activation energies of both the forward and reverse reactions, and change the rate constants to the same extent. Therefore the use of a catalyst to increase the rate of achieving equilibrium does NOT alter the equilibrium constant or the equilibrium composition.