How do I implement the Add trait for a reference to a struct?

You need to implement Add on &Vector rather than on Vector.

impl<'a, 'b> Add<&'b Vector> for &'a Vector {
    type Output = Vector;

    fn add(self, other: &'b Vector) -> Vector {
        Vector {
            x: self.x + other.x,
            y: self.y + other.y,
        }
    }
}

In its definition, Add::add always takes self by value. But references are types like any other¹, so they can implement traits too. When a trait is implemented on a reference type, the type of self is a reference; the reference is passed by value. Normally, passing by value in Rust implies transferring ownership, but when references are passed by value, they’re simply copied (or reborrowed/moved if it’s a mutable reference), and that doesn’t transfer ownership of the referent (because a reference doesn’t own its referent in the first place). Considering all this, it makes sense for Add::add (and many other operators) to take self by value: if you need to take ownership of the operands, you can implement Add on structs/enums directly, and if you don’t, you can implement Add on references.

Here, self is of type &'a Vector, because that’s the type we’re implementing Add on.

Note that I also specified the RHS type parameter with a different lifetime to emphasize the fact that the lifetimes of the two input parameters are unrelated.

¹ Actually, reference types are special in that you can implement traits for references to types defined in your crate (i.e. if you’re allowed to implement a trait for T, then you’re also allowed to implement it for &T). &mut T and Box<T> have the same behavior, but that’s not true in general for U<T> where U is not defined in the same crate.