The basic problem is that when one part of the system shuts down, it will change the loads in other parts of the system (generally, but not always, increasing them). Ideally, the shutdowns would occur in such a way to minimize disruption, but sometimes things don't work out that way.
As a simple example, imagine that you have two communities, each with a power plant, and a link between the two communities. The plants are designed to produce 100 units of power each and have a instant-shutoff limit of 105 units. The link is designed to transport 20 units of electricity, with an instant-shutoff limit of 25.
Suppose community #1 is drawing 95 units and community #2 is drawing 100. Generator #2 shuts down, putting all its load on generator #1.
If the link between the two communities were to drop at this point, community #1 would have power and #2 wouldn't. What could happen instead, though, would be for the link to get all the power generator #1 is producing that isn't being drawn by community #1 (i.e. 10 units), pushing generator #1 past the instant-shutdown threshhold. The link is not overtaxed (it only has 10 units flowing through it), so it remains active. The effect is that both communities lose power even though a different shutdown sequence would have only affected one.
In the real world, there are a many more generators, communities, and links, but the same sorts of scenarios can occur. Although most problems would be the result of generators or links getting overtaxed, problems can also occur if a generator suddenly finds itself underloaded. If a link trips which had been connecting a generator to most of its load, that can cause the generator to run overspeed until a safety trip shuts it down. At that point, whatever part of the generator's load was still connected will either be in the dark or will draw power from somewhere else.
I know there are systems that are supposed to minimize the effects of cascading failures, but breakers don't always trip in the desired sequence.