In particular, move is important if there is something like a unique_ptr. To make a copy, I have to make a deep copy of whatever the unique_ptr points to, which could be very expensive. To do a move, I just copy the bits of the unique_ptr, but now the original object can't be the one that owns what's pointed to.

Sure. Notice std::unique_ptr<T> is roughly equivalent to Rust's Option<Box<T>>

The C++ "move" is basically Rust's core::mem::take - we don't just move the T from inside our box, we have to also replace it, in this case with the default, None, and in C++ our std::unique_ptr now has no object inside it.

But while Rust can carefully move things which don't have a default, C++ has to have some "hollow" moved-from state because it doesn't have destructive move.