Why did the bulbs in my recessed lights burn out?
First you need to verify that you are getting power to the fixture. Sometimes when lamps fail they short and may trip the breaker or damage the dimmer. I have found many lamps that buzz when on a dimmer and it was because the wave shaping of the dimmer caused the filament to vibrate and created a noise. I would verify with a new lamp first and if no light then I would check the breaker  by turning it off and then back on - this will reset it if it tripped but the handle did not move enough to be obvious.

Beyond this, a meter or non-contact tester will be needed. Check for voltage before and after the switch - when turned on both should show voltage. If the switch failed, replace it. Last will be to verify the connections in the light fixtures. Make sure the wirenuts are tight and no broken wires. I left checking the fixture for last because with lighting circuits I have found a bad switch or tripped breaker to be the problem more often than a bad connection.