This book is a great resource. See pdf page 599, actual page 567.
You should go to the page before reading on and while reading the rest of the post.
In it, it explains everything very coherently and breaks down the derivation into 4 steps: finding an equation for the location of the center of the circle (x and y coordinates), and then finding the equation for the point P in in reference to the center.
We will start off by trying to find where the center of the circle is at angle $\theta$. The x coordinate is going to be equal to the distance traveled, which is the same thing as the length of the sector of the circle we have already covered. The sector is equal to the radius times the central angle, so the center will be at $x = a \theta$
The y coordinate of the center at any time is really easy because the center is always the height of the radius, which is $a$. Therefore, the center is at coordinates $(a\theta, a)$ at angle $\theta$.
Now, let's try and find the location of point P in reference to the center. We will start with the x coordinate.
At angle $\theta$, P will start by lagging behind, then jumping ahead, then going back to where it started. Therefore, we want to start by subtracting $0a$, then $1a$, then $0a$, then -$1a$, then going back to $0$ again. This behavior is exhibited by $a \sin \theta$, so our x coordinate is now complete: $x = a\theta - a \sin \theta = a(\theta - \sin \theta)$
Now for the y coordinate. To get the height of point P at angle $\theta$, we notice that it starts out below the center, then goes above the center, then back below. Therefore, we want to subtract $1a$, then $0a$, then $-1a$ (add $1a$), then go back to $0a$ again. The pattern of $(1, 0, -1, 0, 1)$ is exhibited by $a \cos \theta$, so we want to subtract this from the center, giving us $y = a - a \cos \theta$ , or $y = a(1 - \cos \theta)$.
Now, we are done. Our two equations are $$x = a(\theta - \sin \theta)$$ $$y = a(1 - \cos \theta)$$.