How can I replace broken CFL bulb in a recessed light fixture?
That type of bulb just snaps in and out with a little pressure. It probably broke because the UV light had weakened the plastic, or because it force was applied in a sideways direction. 

To remove it, turn off the light via the switch or circuit breaker. Grab the remaining plastic with a needle-nose plier and pull it straight outward. Do not twist or tilt or you risk damage to the socket.