The origin of the language: gingerBill got drunk one night and wrote a compiler. Seriously though, it was an attempt to create a language that would be a reasonable C alternative - simple semantics, usable for performance-critical applications (mainly games and engines), good support for tooling. I would also describe it close to something with underlying mechanics of C combined with the niceness of Go. Odin does improve on Go in a lot of places.
The use cases would be similar to C or C++. There's not much different in the way of memory management / the way objects are stored in memory etc. There is a concept of an allocator which could make Odin a better candidate for good performance when you allocate memory (i.e. putting everything on heap can be slower than bump allocators)
Some of the coolest features I've found using the language - bitsets (make a set of flags out of an enum), slices, which C didn't have but are so nice to use, the or_return, or_continue and or_break operator (hello, Go programmers, tired of writing if err != nil yet?).
Language comparisons:
unlike C: Has simpler syntax, easy to parse, has slices, bitsets, sized enums, templates.
unlike C++: doesn't have operator overloading, has simpler semantics, the standard library is readable.
unlike Rust: no borrow checker, limited lifetime checking (can detect using stack values that have dieded), no lifetime annotations, limited support for typeclasses, no pattern matching, no traits. I wouldn't compare it to Rust, honestly, it's a different kind of language.
unlike Go: I haven't used Go in a while so forgot a few details, don't quote me on these: it doesn't have var syntax for declaring values, you can not shadow tuple declarations at all, support for or_return syntax, tuple returns can be used to saturate arguments of another function, doesn't have garbage collection. Doesn't have a package manager.