Is it low current or low voltage that dims a bulb? At times we have dim bulbs due to some fault from national grid, what I want to know is if this fault is low current or low voltage. 
Other times electrical appliances blow-up, is that high current or high voltage?
 A: In short it's low voltage.
Current, voltage and power are all linked via the impedance (like resistance) of the thing in question.
The voltage is a property of the grid, with something like a light bulb you can assume the grid voltage won't change. it might if you were running an industrial induction furnace maybe, but not with a light bulb. The current drawn will drop as well, but as a consequence of the voltage drop.
What you are seeing of course is a drop in power $P = UI$.
A: Incandescent bulbs appear dim because the filament's temperature is below it's design temperature.
It's running cool because the current flowing through it is lower than normal. Current + resistance = heat for most non-exotic conductive materials.
The current is lower than normal because the applied voltage is lower than normal. As a light bulb's resistance is basically fixed, current will track voltage rather nicely.
So, it's dimming because of low current which in turn is caused by low voltage.
The explosion part can be for a number of reasons. A high voltage spike can cause arcing, breakdown of insulation for any number of reasons (like overheating) can cause a short circuit. My university once had a total power failure because a crow landed on top of the substation. The crow's wingspan nicely matched the input pylons and they had to replace about half the substation. The crow didn't make that mistake again.
