Cold water is more dense than hot water, and therefore thermalizes more fast neutrons (i.e. slows them down). Thermal neutrons are more likely to interact with the fuel, therefore the reactivity added by the cold water is positive. Not all reactors are susceptible to cold water accidents - but of course not all reactors use water as a coolant/moderator. However, the potential exists for a cold water accident on pressurized water reactors (PWRs) with a negative temperature coefficient of reactivity.
The scenario offered in the writeup above is bogus, however. In a loss of coolant accident, the reactor ought normally be scrammed - that is, all the control rods will be fully inserted into the core. Control rods absorb neutrons, so that they are unable to react with the fuel. So a scram will result in adding huge amounts of negative reactivity to the core - so much that adding cold water cannot return the core to criticality, even room temperature water. The water that is normally charged to the primary coolant system to make up for losses is at room temperature and does not cause a cold water accident when it is added, although it is added at a slower rate than is likely for make up water during a loss of coolant casualty.
'Cold' is a relative term. In order to generate higher steam pressures (the steam will be at saturation pressure for whatever temperature it is at), PWRs operate with a high bulk coolant temperature. So, to the core, 400F (204C) may be rather cold, depending on its design.
A more likely scenario would be as follows: A protective action has occured (i.e. control rods have moved in automatically to prevent an accident when dangerous conditions were detected). The cause of the protective action has been corrected, and it is desired to restore the reactor to criticality. During this time, steam demand remains high, causing a large delta T (temperature difference) between the hot and cold legs of the coolant loop. The operator shims out on the control rods to reach a high start-up rate. Once a high start-up rate is established, the boob of an operator does something extremely dumb, like shift the coolant pumps to high speed, raising the rate at which the coolant in the cold leg is passed through the reactor, adding excess reactivity. This will cause a cold water accident, particularly if reactor power is below the point of adding heat.
The thing that is so bad about cold water accidents is that they have the potential to cause the reactor to go 'prompt critical'. This is when enough prompt (not delayed) neutrons are released in a nuclear reaction to sustain the reaction without the contribution of delayed neutrons. This is a very bad thing. Once the reactor has reached prompt criticality, it can no longer be controlled. Power rises at an exponential rate, until so many neutrons are being released in the core that scramming the reactor will be ineffective at shutting it down. A steam explosion could result. Less dramatically, the core may just get hotter and hotter until it melts down. And by down, I mean literally down, because it will probably become hot enough to melt through its containment vessel and start burying itself into the ground or whatever. Not good. This is a 'nuclear melt down' and occured at Chernobyl. There was also an explosion that sent the vessel head through the roof of the secondary containment building, and allowed the release of fission products to the environment.
I'll stop here for now, to avoid ranting beyond the scope of this article. Many of the casualties listed also have the potential to cause prompt criticality and/or a cold water casualty. Let me just say that the prospect of prompt criticality is most terrifying to an operator, and also that many, many protective features and interlocks exist on a well-designed nuclear power plant to prevent it from occuring.