Based off of that video, it looks like that is either a damaged or faulty screen, data cable between the screen/motherboard, or (unlikely but possible) motherboard damage on some part of how it controls/sends video signals.
Potentially a power failure of some kind? I am assuming the computer is still running/fans still spinning/etc. after the screen fades to black?
Can you plug it into an external monitor? It has both VGA and HDMI on the left hand side, so you have a couple of options. If you get a clear and working image on a secondary display, then it is for sure the screen or cable, and worth replacing to repair.
This guide (Sony VAIO VPC-EB33FM Display Replacement) has the process for replacing the screen. You would only need to go so far as step 30, and the cable with yellow tape in that step is the cable that may need to be replaced instead of and/or alongside the screen. The steps after 30 show you how to remove the other components for replacing the entire LCD cover, but that is not necessary to replace the screen.
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