As j has pointed out, the question is a bit vague in the context of what exactly you want to understand, because there is always a lot going on under the hood. :)
Assuming, that you want to understand the workflow, let me explain that in brief, in the context of react-router, because again, React per se doesn't provide any routing utility.
With react-router you map location pathnames to components. Whenever a location is matched to a desired pathname, the corresponding React component (generally, without requiring a new "view" fetch) is rendered in the view, as simple as that!
Other front-end routing solutions work quite similarly! Probably a full blown guide shall help you understand, and appreciate the internals more!