You need to understand the bits you are trying to prove, but not the full proof. It's more like reading haskell types than math, even though the vocabulary is heavily inspired by math.
Is there benefit of using this branded type over just encapsulating the raw string in a private variable in closure or class? This feels a bit like forced nominal typing. The Email type doesn't have to be a string, it can be encapsulated so that invalid Emails are not representable.
There are codebases out there with enormous amounts of duplication, filled with implicit dependencies. You just haven't encountered them to appreciate good abstraction.
The author has huge online following, and influence over developers. I don't doubt the author is genuinely interested in this, but the employer definitely is supporting this too.
The spec and proof are separate. In this blog article he mentions seL4 formal verification, where they state that the spec was 4900 lines of Isabelle and the proof was 200K lines. Obviously human has to understand the spec deeply.