It's a standard crappy switcher.
Do you have any experience with measuring voltages at mains potential?
The mains input is rectified, the two electrolytics on the right hand side are the primary bulk capacitors.
At first, I'd ask you plug it in, turn it on and measure the voltage across these - it's either near zero or some 320V (in a 230V country) or 160V (in a 110V country). If you don't know what I mean, stop here - it's just dangerous. If it's zero, check the fuse, rectifier diodes and other parts on the way from mains input to the capacitors.
Next stage is a (most probably) MOSFET half bridge and the high frequency transformer (the large part with yellow tape in the middle). The MOSFETs chop the rectified mains to some 10kHz, this goes through the transfomer.
The primary MOSFETs are mounted at the right hand heat sink. One could check them with a DMM in diode mode, but I suspect they're OK, because they often fail shorted and let the smoke out. This would be visible here.
Mounted on the other heatsink are rectifier diodes, these rectify the high frequency into the output DC voltage. The toroid is the main storage inductor for "smoothing" the output.
The IC (a standard TL494) is the control circuitry and regulates the output voltage. It's located on the secondary side, so this circuit must have some self oscillating capabilities to enable start up.
You most probably won't find a fault here without a scope, as you'd have to watch if the control circuitry tries to start up.
Did I mention it's quite dangerous to work with DMM and scope on a mains powered circuit? I you're unsure and don't know what I'm speaking of, better stop here.
Some other method would be to test the components in a "safe" (non-powered) state and try to find a defective component. Check the discrete semiconductors, the electrolytics and the red film capacitor. Transformers and inductor typically don't fail here. Maybe you'll find an open resistor or a bad solder joint. Fix what you found, and carefully test the unit. It might explode on you.
If you google for "PC silver box" schematics of the older types, you'll find diagrams of power supplies that work in a similar way.
TBH, I'd just bin the innards and re-use the case. It's a crappy supply.