Why can geometric figures, such as a straight line move?

You are right that a point (or other figure) by itself cannot move - this is an abuse of language. Instead of

Let $Z$ be the point $X$ rotated around $Y$ by angle $\alpha$

a more precise formulation would be

Let $Z$ be the point that is uniquely determined by the conditions $|YX|=|YZ|$ and $\angle XYZ=\alpha$

and similarly for figures instead of a point $X$, or with reflections, translations, or whatnot instead of rotations. $Z$ is not an actually moved version of $X$, but rather the result of a mapping applied to $X$. As "wrong" as it is, the first formulation is really way more intuitive.

The same problem arises outside of geometry when we say, for example "pick a number and double it".