Bad capacitors. It's always bad capacitors (and it's never Lupus).
They've dried out and become resistive. In the process they sometimes make life ugly for power transistors and diodes.
If you're not safe with some mains voltage (there may be some in the hot side of the supply) please don't open the box.
Look for swollen electrolytic capacitors (metal cans). Replace with reputable parts with same values (no consumer components supplier will sell parts as nasty as a cheap manufacturer can buy wholesale).
Check diodes and transistors on output side for blowage.
Main high voltage mains side capacitor might be dried out. Shows up as overload in feedback paths and shutdown. Replace it. Must be same voltage rating!
Go see the sci.electronics.repair FAQ.
Try to avoid contention
It's not a good idea to put the two outputs of the PSU into contention like this (shorting the outputs of the 5V and 5V-SB together).
When the PSU is off, you are back-feeding the output of the 5V supply which will draw current from the standby supply reducing the available useful current and possibly damage it over time. It may potentially draw sufficient current to short the output or dramatically reduce its output voltage.
When the PSU is running, you are putting two independent outputs in contention where they will fight for the exact final voltage of the line (also bad for efficiency/longevity). In some ATX supplies the 5V-standby is linearly regulated while the main 5V is switched. This can result in current spikes in the weak linear regulator resulting in failure.
Two simple solutions
You need to isolate the two supplies. Either use a wired-OR (two diodes, one on each 5V output, in a common-cathode configuration) or run your MCU from the standby supply full-time (if possible).
The more advanced solution
There are also active circuits (microchips/IC's) that are optimized for 5V standby power supply switching.
Various wired-OR configurations:
LTC4411
LTC4413
Power switches:
MAX14525
MAX1823B
Be careful with some of these power switches. For example, with the Max1823b you will need to make sure that you add logic to prevent both outputs from turning on at the same time.
Note: When following my links, click on "documents" to see datasheets and app notes.
Best Answer
It's a filter. The big round thing is a common mode choke. It cleans up the incoming mains and prevents the high frequency noise generated by the switch-mode power supply (SMPS) from getting into the mains AC.