Rust is a programming language. It's a general programming language, so it's good for a large variety of things. Different people have different reasons for using Rust. There are three large constituencies, as I see them: systems programmers, functional programmers, and scripting language programmers.
Before I get into details: all generalizations are false ;)
Systems programmers come to Rust because it's able to do the lowest levels of software: operating systems, device drivers, stuff like that. Where it improves on existing systems languages is "safety." The Rust compiler does a lot of compile-time checking to make sure you don't do things wrong. Existing languages force you, as the programmer, to double check your work. We have the compiler double-check your work, and force you to get it right.
Functional programmers see all that static checking, and it feels like home. But a home where they have significant speed gained. Functional languages can be fast, but not always C level fast. Rust is. It's easier to get that level of performance out of Rust.
Scripting language programmers come to Rust for two reasons: the first of which is to write extensions to their language for speed. You can write a Ruby gem in Rust, rather than in C. This is useful because this low-level stuff is not their expertise, and so the extra checks are really nice. Related: we have a lot of the nice tooling that scripting language folks have come to expect from their language. No need to write makefiles, we have Cargo. The second is sort of related, but these checks also mean it's easier to learn low-level stuff, so if they want to expand their universe, Rust is a nice way to do so. We're seeing a lot of people for whom Rust is their first systems language.
I didn't know where to put this one, but Rust focuses on a concept called "zero-cost abstractions." This means that a lot of our features that feel very high level are extremely efficient. You don't write manual for loops, you se iterators. You don't deal with pointers directly, but with references. Structures like Box and Vec do memory management, but you don't need a garbage collector, and you don't need to manually manage it, but you also don't need to explicitly call free().
These are very broad strokes. Lots of other people have other reasons too. But that's some of them.