In an inertial frame of reference, it doesn't. If it's an ideal pendulum, the pendulum goes on swinging exactly as it did before. Even realistic pendulums are only slightly distorted by the effect of friction and air resistance.
But we are standing on the Earth, which is not an inertial frame of reference. The rotation of the Earth under the pendulum makes the pendulum appear to change direction from our frame of reference. An apparent force (or "fictitious force") emerges, the Coriolis force, which is not a fundamental force like gravity, but a result of using a rotating frame of reference. This force is proportional to the velocity of the pendulum and the rotation of the Earth, and acts in a direction perpendicular to the pendulum's motion.
This force is very small, because the Earth's rotation is not very fast (2pi radians per day by definition). But it can be measured, and if you leave a good pendulum swinging back and forth over the course of a day and mark its movements you can see how the motion will precess around in a circle.