It's somewhat subjective, but the core of Typescript which AssemblyScript uses for 'inspiration' is a much smaller and 'friendlier' language than C++ or Rust. Unlike C++ or Rust you can learn the whole language in an afternoon or at most a weekend, and if you already know TypeScript anyway you also already know AssemblyScript.
Basically, the question "when targeting WASM, why should I use C++ and Rust instead of AssemblyScript" is just as valid.
In the end the language choice is most likely enforced by what libraries one needs to use for a specific project anyway (e.g. if you need to build a project on top of a handful C++ libraries, it's probably a good idea to use C++ for the whole project too - and at least there is such a choice now when targeting the web).
Basically, the question "when targeting WASM, why should I use C++ and Rust instead of AssemblyScript" is just as valid.
In the end the language choice is most likely enforced by what libraries one needs to use for a specific project anyway (e.g. if you need to build a project on top of a handful C++ libraries, it's probably a good idea to use C++ for the whole project too - and at least there is such a choice now when targeting the web).