https://verus-lang.github.io/verus/publications-and-projects... - The papers here go into their implementation. They take the proof statements and information about the program and turn it into an SMT problem (and run it through Z3 if I read correctly) and then they use that to prove the properties of the program.

SPARK/Ada and Dafny work similarly, and have good documentation if you want to try your hand at something with a (presently) better set of documentation.

https://mitpress.mit.edu/9780262546232/program-proofs/ - Dafny book, pretty good tutorial on the topic

https://learn.adacore.com/courses/intro-to-spark/chapters/01... - Free tutorial for SPARK

All it does is dispatch proof obligations to an SMT solver like Z3. There is nothing special about Verus, it works in the same way other program verification frameworks like Dafny and Frama-C work - except it's for Rust. Most of this article presents nothing unique to Verus and is more of an advertisement for the authors research work and the other work AWS is doing.