The traits each represent more and more restrictive properties about closures/functions, indicated by the signatures of their call_... method, and particularly the type of self:
FnOnce (self) are functions that can be called once
FnMut (&mut self) are functions that can be called if they have &mut access to their environment
Fn (&self) are functions that can be called if they only have & access to their environment
A closure |...| ... will automatically implement as many of those as it can.
- All closures implement
FnOnce: a closure that can't be called once doesn't deserve the name. Note that if a closure only implements FnOnce, it can be called only once.
- Closures that don't move out of their captures implement
FnMut, allowing them to be called more than once (if there is unaliased access to the function object).
- Closures that don't need unique/mutable access to their captures implement
Fn, allowing them to be called essentially everywhere.
These restrictions follow directly from the type of self and the "desugaring" of closures into structs; described in my blog post Finding Closure in Rust.
For information on closures, see Closures: Anonymous Functions that Can Capture Their Environment in The Rust Programming Language.