Why do these formulas rotate a point $(x,y)$ counterclockwise or clockwise by an angle of $\theta$? I have no idea how to start; I want a step-by-step explanation.
Counterclockwise: $x'=x\cos\theta-y\sin\theta$, $y'=x\sin\theta+y\cos\theta$
Clockwise: $x'=x\cos\theta+y\sin\theta$, $y'=-x\sin\theta+y\cos\theta$
See image. This represents a counter-clockwise rotation of an original red segment (terminating at the original point) to a new blue segment (terminating at the new point). The line segment length $r$ is held constant.
The original coordinates $(x,y) = (r\cos\theta, r\sin\theta)$
The new coordinates $(x',y') = (r\cos(\theta + \alpha), r\sin(\theta + \alpha))$
Using angle sum formula, let's re-express the coordinate $x'$:
$r\cos(\theta + \alpha) = r\cos\theta\cos\alpha - r\sin\theta\sin\alpha = x\cos\theta - y\sin\theta$
You can now work out $y'$ and handle the clockwise case yourself.