Why my LCD monitor backlight turns off?

"Looking for picture behind dark screen

If you can see picture even while backlights are not working then problem almost certainly lies in monitor powersupply/inverter board.

Sometimes it will be extremely hard to see picture when backlights are gone. You should try external lighting at different angles and at different luminosity. If possible, it could help if there is fullscreen animation running while checking this, for example cartoons with great contrast and bright colors.

Causes of darkness

Blown/leaking/dry capacitors are very common problem with flat screens, they are also pretty easy to check for defects (most of time visual inspection is all you need).

Capacitors are also relatively cheap and pretty easy to replace (if you know how to use soldering iron), just remember to check orientation for +/- leads (likely marked on board). However you should also inspect those high voltage cables for burn marks, especially near cold cathode tubes (not found from screens with LED backlights).

More on topic

On laptops, there's small inverter board located just under screen frame casing (in most models) that is often cause of black screen. These could be very cheap for some models.
See PSU failing or Mainboard failing? for picture of defective capacitors.
There's also simple power cord test that you might want to carry out:

Pressing power button (pulling plug out for few seconds) many times also speaks toward broken PSU case as
when you press and release button it will discharge capasitors and
then immediately charge them again, doing this many times could still
make almost broken capasitor to work once again for some moments."