This is crosspost of this MSE question.

Having carefully read the proof of Beck's monadicity theorems and some related variations, I'm now hungry for cool applications.

For instance, I found these blackboard pictures of a lecture by David Jordan, but I do not understand any of the examples - too terse for me.

What are some interesting applications of monadicity theorems, especially in geometric contexts (algebraic geometry would be especially cool)? Can anyone spell out the example(s) presented in the linked pictures?

**Added.** I know about monadic descent and am trying to slowly understand it. I am hoping for additional examples.