DIAGNOSIS: This is a common handshake issue between the monitor and your PC's power management. The PC sends a wake signal, but the monitor doesn't recognize it and stays in standby.
FIX: Start with the simplest fix. 1) Power off the monitor with its physical button. 2) Unplug the DisplayPort or HDMI cable from both the monitor and your PC. 3) Unplug the monitor's power cord from the outlet for 60 seconds. This performs a full power cycle and clears the monitor's internal memory. 4) Plug everything back in firmly and power on. This often resets the communication.
IF THAT DIDN'T WORK: The problem might be on the PC side. Go into your Windows Power Options (search for it in the Start menu). Click 'Change plan settings' for your current plan, then 'Change advanced power settings.' Expand 'PCI Express' and set 'Link State Power Management' to 'Off.' This prevents your graphics card from going into too deep a sleep.
HEADS UP: Also check your graphics card driver software. Both NVIDIA and AMD have settings related to monitor power and detection; make sure nothing is set to aggressively turn off the display output.