A Cartesian plane is just a set of points. Among those points are some that constitute a straight line. Let $L$ denote such a set of points that constitute a line and $A$, $B$ denote two distinct points on that line. Then, $AB$ denotes a segment of the line. Now, we have heard this expression "let $AB$ rotate about its one endpoint, say $A$" many times. Saying that, we would then go on to draw another line segment $AB_1$. And we would say, the line segment $AB$ has rotated from its initial position ($L$) to its final position $L_1$, the line that has both the points $A$ and $B_1$.
My question is, how can a line, a set of points, move on a plane? To understand that better, let us consider a single point first. If a point that constitute a plane moves on that plane, what does that really mean? If it moves from its initial position, then it's no longer there. But if it's not in its position, then it must be located on someone else's position, assuming there exists a point at any location on a plane. It then follows to say, there are more then two points that can exist at a certain location on a plane.
What exactly are points? If it moves from its initial position, is there a point at that position any more?
You are right that a point (or other figure) by itself cannot move - this is an abuse of language. Instead of
a more precise formulation would be
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".