The reason is there are always forces which resist the motion of the oscillator. Due to this effect, the oscillator continuously works against these forces. As a result, the energy of the oscillator is continuously dissipated against these resistive forces which decreases the amplitude of oscillation.
As an example, consider the motion of a simple pendulum. As it moves, it suffers resistance from the air in its way, or friction at the pivot (point where it is suspended). The air resistance continuously decreases its K.E by decreasing its speed. In this way the P.E also decreases as during the course of motion of the pendulum, the K.E is converting to the P.E and vice versa. This gradually decreases its amplitude in every trip and finally it comes to stop.