In the example images, how did the red area and blue area equal the green area?

That's the trick. It's hard to tell directly that red + blue = green. If you could, the Pythagorean Theorem would be obvious. The areas of both big squares are equal to (a+b)^2. If you take away 2ab in the form of two a by b rectangles, you are left with the red and blue squares shown in the left diagram. If you take away 2ab in the form of four a by b right triangles, you are left with the green square in the diagram on the right. If you start with equal areas, and subtract equal areas, what you are left with must have equal areas.