Eli5: Why do nuclear plants need external power? Couldn’t they just siphon off some of the power the plant makes to accomplish the same tasks (like an alternator running the electrical components in a car)?
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Yes, and they do that. However if you shut off the reactor, you can’t use its power output. The core, fresh out of neutron bombardment, still generates enough decay heat to melt itself into a puddle on the bottom on the reactor vessel though (this occurence is known to lead to many cups of coffee being consumed and frequent swearing).
So, you need to get electricity from *somewhere* to run the cooling pumps. Usually you just grab power from the main network, but just in case there is an outage, nuclear power plants have several diesel generators ready to do that, just make sure you don’t run out of fuel as well before the grid gets unfucked.
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