Who's this everyone person you're talking about? Any exercise will tear down/damage muscle mass if the exercise is intense enough, but cycling "eating away" at fat and muscle...makes it sound like such a negative thing.

True, cycling will use fat as a substrate for energy production, so that could be considered "eating away" at the fat. At low intensities, your body will preferentially use fat as a fuel, but that use of fat will diminish more as you increase the intensity. With long, especially ultra-endurance, events, you will have a breakdown of muscle mass (i.e. protein contributing to energy production) due to damage and fuel utilization.

The actual breaking down of muscle is good because as long as you're recovering properly, you will build and repair muscle...this is what makes you stronger, bigger, etc. But if you're only cycling, it's highly doubtful that you'll see an increase in muscle mass, or even a large loss in muscle mass due to damage or tearing down of muscle, because of the nature of the sport. To really produce more muscle, strength train in the gym.