The code-generation phase of the Rust compiler is mainly done by LLVM. LLVM is a set of tools for building a compiler, most notably used by the C[++] Compiler clang[++].
First, the Rust compiler (just like clang, for example) does all the Rust specific stuff like type and borrow checking; in the end, it generates LLVM-IR. IR stands for intermediate representation and it’s… comparable to assembly, but a tiny bit more high level and most importantly: platform independent. Then the Rust compiler just calls ☏ LLVM and says:
Hey buddy, could you please take this IR and generate machine code for the current platform? That would be fantastic ◕ ◡ ◕
To which LLVM responds:
🌈 Sure, no problem, new friend. Here is your highly optimized machine code for [e.g.]
x86_64! ♫♪♫
Afterwards they invite a few more friends to wrap it all up in a nice little [e.g.] ELF package and beautifully place it in the users file system. (and the user is like…)
Information like this can be found in the official FAQ which contains a lot of interesting information anyway. For more in-depth details on the Rust compiler, you can read the “Rustc Guide”. For this question, the chapter “High Level Overview” is pretty interesting.