Identifying translations and rotations as compositions. I am having trouble understanding the below which are the ones underline in red and blue.

For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$
As for the blue: Why is that $R_{A,90}(B)=A'$

The transformation $R_{A,90^\circ}$ is a counterclockwise rotation of the plane through an angle of $90^\circ$ around the point $A$. Since $A$ is the centre of rotation, it doesn't move, and therefore $R_{A,90^\circ}(A)=A$. When $B$ is rotated $90^\circ$ counterclockwise around the point $A$, it ends of at $A'$: it started directly below $A$, so it ends up the same distance directly to the right of $A$, at $A'$. Think of it as an hour hand moving backwards $90^\circ$ from $6$ o'clock to $3$ o'clock.
The transformation $\tau_{AB}$ is a translation of the whole plane in the direction from $A$ to $B$; the length of the translation is the distance $d$ from $A$ to $B$. If you translate (move) the point $A$ $d$ units directly towards $B$, it ends up at $B$, so $\tau_{AB}(A)=B$. Similarly, the point $Y$ ends up at $X$, because $|YX|=|AB|=d$, and $X$ is directly below $Y$, so that the lines $YX$ and $AB$ are parallel.
