Mosquito Trap high Voltage
The plates could be deadly or cause injury if you touch them, depending on what else you are touching (such as a ground) and whether you touch one or both plates.

Usually safety agencies require that a proxy for a human (child human) finger  should not be able to touch anything harmful. Here is an articulated test "finger" for IEC testing from this website. Rather expensive and specialized device, but you get the idea. 

 

It looks from the photo like the gaps in the housing may be too large and the electrically "hot" parts too close to the face to be safe. There should also be a fuse -- maybe you didn't show it since you didn't show the switch. 

There's not necessarily much inherently unsafe about this circuit but it would have to be protected, and the housing needs to be  made from appropriately fire retardant plastic. All of which should be doubted without evidence to the contrary. 

Aside from safety, I have my doubts that the relatively low 600V voltage will actually kill bugs- effective bug  killers usually have a step-up into the kV. Not all bugs are that 'juicy'.  It might just tickle them a bit. The visible LED should either be inverse parallel LEDs or have a parallel shunt diode.  



As to how it works  (to the extent it actually does work), the two diodes and 220nF capacitors form a voltage doubler, which will produce around 660V peak across the plates with no load. It is effectively short-circuit proof (barring failures of the capacitors or diodes) because a direct short between the plates will only conduct current through the two 220nF capacitors in parallel (440nF), which is about about 33mA assuming 240V/50Hz. However if one of the parts fails there should be a fuse to limit the current and prevent a fire. The LED (see my note above) appears to use a capacitive dropping circuit but the values you have found may not be correct. There is a bleeder resistor across the cap to prevent the pins from giving the user a tingle when they are pulled out from the wall.