How does multiplying by trigonometric functions in a matrix transform the matrix?

The matrix $$\left[\begin{align} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{align} \right]$$ when it acts on a vector it rotates the vector by $\theta$ in clockwise direction. Hence when $\theta = 90^{\circ}$, it rotates the vector $$\left[\begin{align} a_1 \\ a_2 \end{align} \right]$$ from vertical to horizontal clockwise.

Well, the rotation matrix they write there is

$$ \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right) $$

If you multiply a vector $\left( \begin{array}{c} a_{1} \\ a_{2} \\ \end{array} \right)$ by this matrix then you end up with $\left( \begin{array}{c} a_{2} \\ -a_{1} \\ \end{array} \right)$ If you draw a picture in the $xy$ plane connecting each of $(a_{1},a_{2})$ and $(a_{2},-a_{1})$ with the origin, it will be clear that the latter is a 90 degree rotation of the former.