Diagnosis
The microwave is getting power to its 'housekeeping' circuits (light, fan) but the high-voltage system that actually creates the microwaves isn't engaging. The most common culprit is a failed door switch.
How to Fix It
1. Unplug the unit. 2. The door switches are activated by little plastic tabs on the door. Open the door and look for them. 3. Carefully inspect the plastic actuator tabs on the door itself. If one is broken or worn down, it won't press the switch. 4. If the tabs look fine, the switch itself is likely bad. You'll need to remove the outer wrapper to access and test the switches with a multimeter.
If That Didn't Work
If all door switches check out, the next suspect is the high-voltage relay on the main control board. It's a small component that sends the 'go' signal to the magnetron. When it fails, everything seems normal except the heating.
Heads Up
Never, ever try to run the microwave with the outer casing removed or the door interlock system bypassed. The radiation exposure is extremely dangerous. If the fix involves the internal switches or high-voltage components, consider professional help.