As the pendulum enters the magnetic field, it notices a change in magnetic flux. The change in magnetic flux induces an electric field, and by Lenz's law the induced electric field will oppose the pendulum's direction of motion. This dampens and stops the pendulum.
Look at the picture in the Berkeley Review passage you most likely got this question from. If you notice, the pendulum is actually part of a conducting loop, and as the pendulum swings back and forth, the area of that loop changes. It happens that the pendulum is swining in such a way that the area of the loop is perpendicular to an external magnetic field, which means that the flux is changing as it swings back and forth. As point out in the post above, a change in flux induces current, which means that energy is being drained from the pendulum. If you rotated the system by 90 degrees (while keeping the B field the same), then you'd have no flux and no dampening.