Diagnosis
You have power getting to the main control board (that's the click you hear), but the board itself isn't booting up or sending power to the display and buttons. This can be from a failed board or, more hopefully, a loose connection.
How to Fix It
First, do a hard reset. 1) Unplug the dryer from the wall outlet. Don't just turn off a switch—physically pull the plug. 2) Wait a full 5 minutes. This allows any residual power in the capacitors to drain, which can sometimes reset a locked-up board. 3) Plug it firmly back in and try powering it on.
If That Didn't Work
The problem is likely internal. Check your home's circuit breaker/fuse for the dryer to ensure it hasn't partially tripped. If power is confirmed, the main control board or the smaller UI (user interface) board behind the display may have failed and needs replacement.
Heads Up
Control board issues are not a DIY fix for most folks. That click means high-voltage parts are engaging, so working inside the machine while plugged in is extremely dangerous. If the hard reset fails, you'll likely need a technician to diagnose which board is faulty.