The website doesn't explain how it works in a lot of detail. I am the author of tsonic [1], a TS compiler that produces binaries via Clr NativeAOT (on Linux/Mac). The hardest parts were numbers (TS has no ints or shorts), Generics, and TS Utility Types. I've been on it for the last 6 months (almost every day); getting to near complete TS compatibility is a very long journey because of its expressiveness.
Add: A request is to explain how it works on the website. I did take a look at https://www.perryts.com/en/internals/ but are those techniques described really sufficient to express TS? Based on my experience, I must say I'm surprised. But the proof is in the pudding, and if it's compiling those examples it must be working somehow!
[1]: https://tsonic.org