The answer to 'why is my recessed lighting not working' is that there could be several common causes, including a tripped circuit breaker, a burned-out bulb, a problem with the light fixture or wiring, or an issue with the dimmer switch (if installed). The symptoms may include the lights not turning on at all, flickering, or only partially illuminating.
To troubleshoot, first check the circuit breaker to ensure it has not tripped. If the breaker is fine, try replacing the bulb. If that doesn't work, you may need to inspect the fixture itself or the wiring connections. Dimmer switches can also fail over time, so that may need to be replaced. For any more complex issues, it's best to call a licensed electrician, as working with electrical systems can be dangerous for DIYers.