Diagnosis
That click is often the sound of a internal safety switch or relay tripping. Combined with the immediate shutoff, this points to a serious electrical fault, like a short circuit or a failed main board.
How to Fix It
Unplug the device immediately. Check the very basics: try a different, known-working wall outlet. Ensure the filter is clean and completely dry (see the first fix). Check the cord and plug for any visible damage. If all that looks good, the problem is internal.
If That Didn't Work
There are no user-serviceable parts for this. Do not attempt to open the handle. This behavior requires professional diagnosis. Your only path is to contact Dyson Support. Be prepared—this is often a full unit replacement if it's out of warranty.
Heads Up
Do not keep plugging it in and trying to start it after it clicks off. You could cause further damage or, in a worst-case scenario, create a fire risk.