That picture says to me that you've got a dead screen, particularly if all the other features of the console are functioning.
What clnsgne said, first step is probably to open it up and make sure there's a strong connection from the ribbon cable, and that the cable is undamaged. this isn't too difficult to do so long as you're careful (and grounded, of course):
- Shut down the console and remove the battery.
- Remove the six case screws on the bottom of the console (these are tri-wing screws so you'll need a appropriate screwdriver)
- Carefully separate the two halves of the bottom shell. You want to be careful here because the shoulder buttons are situated with springs, which could go flying if you're rough and get lost. Fortunately those buttons and their springs are in the bottom half of the bottom shell, which does not contain the main board itself. So you should be able to separate the two halves without even having to touch the shoulder button assemblies and set them aside.
- Remove the three screws from the main board (should be standard philips this time).
- The main board should only be attached to the rest of the console by the one ribbon cable to the screen, at this point. Lift it out carefully, swinging it like a door, so as not to put pressure on the ribbon cable. Note that some of the buttons and their rubber pads might fall out when you lift the board, so take care not to loose them. These are easy to replace if they do fall out, because they're keyed for each hole they go into, and their pads fit snugly on top of them.
Mind the very, very tiny screws on the SP, that are easily stripped if you're too rough.
Now you can check to see if the cable is undamaged/well-seated. If it needs to be re-seated, take care not to damage the tiny plastic prongs to either side of the port for the cable that are intended to lock it in place. While you're in there, it couldn't hurt to do some dusting and general, electronics-appropriate cleaning.
If none of this helps, you probably have a bad screen. Which...well, there's not much you can do about that except get an entirely new console, find a new replacement screen online, or harvest one out of another GBA SP. It's not impossible to replace these, but since you're working with the hinge now, it's more complicated than the above.
(Of course, do all of this at your own risk. I'm not specifically advising you to do anything - there's always a chance you could make things worse with your console if you start digging around in it, so if you're not comfortable, don't do it. Also I haven't taken apart a GBA SP in some time - this is just what I can remember about the process.)
Hope that's helpful in some way!
