Diagnosis
This is almost always a power issue with the monitor unit itself. The camera's light being on confirms the camera is getting power, but the handheld monitor isn't.
How to Fix It
First, unplug the monitor's power cable from the back of the unit. Then, unplug the small barrel connector end from the cable itself. Wait a full 30 seconds. Now, plug the barrel connector firmly back into the cable, then plug the cable back into the monitor. Finally, plug the wall adapter back into the outlet. This full power cycle clears a lot of glitches.
If That Didn't Work
Try a different USB power cable and wall adapter if you have one. The original cables can wear out where they bend. The monitor needs a solid 5V connection.
Heads Up
Avoid using the monitor while it's plugged in and charging if the battery is completely dead. Let it charge for 20 minutes first, or it might behave unpredictably.