Once a certain damage threshold has been reached, unlike other objects, it won't immediately be destroyed. It will instead flash for 5 seconds before exploding. However if it is dealt enough damage before the 5 seconds is up, it will explode earlier.
If you don't have a wrench, you character will say, "I have nothing to tamper this with!"
It is entirely possibly, given your character is strong enough, to blow up a generator immediately with a single melee hit (bypassing the countdown mechanic entirely).
Despite seeming to be generating power for the facility it's in, when destroyed it doesn't switch off other powered objects such as security cameras. In fact there is little effect whatsoever, besides the explosion itself and making noise.