From the Rust book's chapter on ownership, non-copyable values can be passed to functions by either transferring ownership or by using a mutable or immutable reference. When you transfer ownership of a value, it can't be used in the original function anymore: you must return it back if you want to. When you pass a reference, you borrow the value and can still use it.
I come from languages where values are immutable by default (Haskell, Idris and the like). As such, I'd probably never think about using references at all. Having the same value in two places looks dangerous (or, at least, awkward) to me. Since references are a feature, there must be a reason to use them.
Are there situations I should force myself to use references? What are those situations and why are they beneficial? Or are they just for convenience and defaulting to passing ownership is fine?