The answer to 'my recessed lighting won't turn on' is that there are several common causes, including a tripped circuit breaker, burned out light bulbs, loose wiring connections, or a problem with the recessed light fixture itself. Symptoms may include the lights not turning on at all or flickering/dimming.
To troubleshoot, first check the circuit breaker to make sure it hasn't tripped. If the breaker is fine, replace any burned out light bulbs. Then inspect the wiring connections in the light fixture and junction box to ensure they are tight and secure. If the issue persists, the problem may be with the recessed light fixture itself, which would likely require replacement by a professional electrician.