He didn't skip anything but he left the details to the reader:

Let C be the area of the big triangle, A and B be the areas of the two small triangles. By construction we know that C = A + B. Moreover, a, b, c are the hypotenuses of the triangles A, B and C.

The area scaling quadratically with the similarity ratio means that

A = (a/c)² C, and B = (b/c)² C.

Now, plug this into A + B = C, cancel C, rearrange.