I don't know how the "self hosted" clojurescript implementations like this and this are implemented.
However, given that the clojurescript compiler is written in clojure and it compiles clojure to javascript, I can reason that the clojurescript transpiler could theoretically transpile it's own source code to javascript, producing a clojurescript transpiler on the browser/node platform. I was just curious, is that feasible and actually how it's done?
Yes, your description sounds fairly accurate.
Here is a post that provides some explanation:
https://blog.fikesfarm.com/posts/2015-07-17-what-is-bootstrapped-clojurescript.html
and a talk that covers some of the same subject, especially near the beginning: