There are two ways these led's are switched on, depending on the board design. Both methods use the +5v supply to power the led. The diagrams show the state when the led would not normally be lit.

The first method is that both anode and cathode are at ground level when the led is off, and the anode voltage is raised to turn the led on.
Code:
5v─────┐
       /────┐
gnd────┘    │ anode
          led
           │ cathode
gnd─────────┘
The second, simpler to implement, method is that both anode and cathode have raised voltage when the led is off (no voltage difference=no current flows so the led stays off), the cathode voltage is dropped to turn the led on.
Code:
5v──────────┐
           │ anode
          led
5v─────┐    │ cathode
       \────┘
gnd────┘
If the board uses the second method then obviously when the board is powered off then both anode and cathode are at ground level.
Code:
gnd─────────┐
           │ anode
          led
gnd────┐    │ cathode
       \────┘
gnd────┘
If the anode is connected to +5v standby then the lamp will light when the board is powered off (since the cathode is now at ground).
Code:
5vsb────────┐
           │ anode
          led
gnd────┐    │ cathode
       \────┘
gnd────┘
However, it will operate normally when the board is powered on.
Code:
5vsb────────┐
           │ anode
          led
5v─────┐    │ cathode
       \────┘
gnd────┘
I can't think of any other scenario which would allow the led to operate normally when powered on.