Q: How did anyone determine the loss of fat and muscle?

A: It can be determined by keeping track of weight and body fat percentage. Body fat can be measured reliably by underwater weighing and somewhat less reliably by a professional with calipers (various scales and devices that send electricity through your body to measure body fat are not reliable at all). If you know starting and ending body weight and body fat %, then you can have a good approximation of fat and muscle lost. It won't be exact, because fat and muscle loss will not be the only changes you experience, but they will be by far the most dominant ones, so the numbers give you a good approximation. For example...

Suppose you weigh 200 pounds at 20% body fat. That means you have 200 * 0.20 = 40 pounds of body fat. After half a year of diet and exercise you weigh 170 pounds at 12% body fat. That means you now have 170 * 0.12 = 20 pounds of fat. So you lost 200 - 170 = 30 pounds total of which 40 - 20 = 20 pounds was fat. That means the other 10 pounds you lost was muscle. This would probably be considered a successful weight loss since you lost significantly more fat than muscle. Again, this is not 100% exact, but gives a good approximation.