Rust uses the LLVM toolchain, which is a a set of programs used to compile LLVM's intermediate representation (IR) into platform-specific code. The process works a little like this:
Rust Code -> Rust Compiler -> LLVM IR -> Back-end -> Platform-specific code
Currently, there is no backend for the Xtensa architecture used by the ESP32. This, unfortunately, means that you cannot compile Rust code for the ESP32. You did say install in your question, but I assume you didn't mean that — rather, you want to run Rust code on the ESP32, not install the Rust compiler there.
If you're really desperate to use Rust, you could, in theory, use the LLVM C Backend, which converts LLVM IR -> C, then use the toolchain provided by Espressif to compile to ESP32 machine code. However, this will be difficult, and might not even work at all. You will find it a lot easier to bite the bullet and use C in this case, which is a shame, because Rust is a great language, but its embedded support is not as good as C's at the moment.