When an LCD display (laptop, notebook, or desktop) suddenly goes black, but one can still see a faint silvery image of text and graphics by shining a flashlight on the screen, that almost always means that the power-supply circuit for the back-lighting lamps has good bad.
The older type of back-lighting lamps were simple white fluorescent light tubes, similar to those you see lighting offices and some kitchens. These lamps are powered by a device known as an "inverter". This device is a long narrow circuit board, typically 15cm long by 1.5cm wide by 0.75cm tall. It usually resides in the upper (screen) half of your laptop, at the bottom, just above the hinge and the power button. A replacement inverter can typically be found for about $25, so it's not an expensive part. You can either buy one and replace it yourself (if you're good with tools, have steady hands, and can find a repair manual for your computer), or you can take it to a repair shop. I suggest the repair shop unless you have prior experience repairing portable computers; they have a lot of small parts which are easy to lose or break.
Newer LCD monitors, however, use arrays of LEDs to back-light the LCD screen. If this is the case with your monitor, the the power-supply circuit for the back-lighting LEDs has probably burned out. I'm not sure what kind of circuit that is, what it's called, or where it's located, but a look at the manual for your computer should reveal something called "LED Power Supply" or words to that effect; that's definitely the part I recommend replacing first. (Or having a technician look at, if you go that route).
In either case, the back-lighting power-supply circuit for a laptop or notebook computer is probably located in the lower part of the upper half, near the hinge and power button, which is why you smelled the burning smell at that location.