Can somebody explain in great detail how to find a and b? ?

So you know that this triangle must be a 45-45-90 degree triangle (you can tell this from the picture!)

The side that is length 4 is opposite from the 45 degree angle, and remember that congruent angles also give you congruent sides. This means that the length of a must be equal to the length of 4, because both are located opposite a 45 degree angle.

To find b, you must know that the hypotenuse of a 45-45-90 triangle is just equal to a. Thus, the length of b= 4 rt. 2