Why is this true?  (cos theta, - sin theta) is what we get if we rotate (1,0) clockwise by theta.  So it is just the x axis, rotated by theta.  Similarly, (sin theta, cos theta) is the y axis rotated clockwise by theta.  Then, x’ is the inner product of the rotated x axis with (x,y).  So it is just the x coordinate of the point according to the new x axis.  And similarly, y’ is the coordinate of (x,y) relative to the new y axis.