Muscle-Ups are difficult to understand in concept. It's a pull-up followed by a dip. Being able to do both pull-ups and dips doesn't automatically allow you to perform muscle-ups.
I intentionally keep strict form with all of the exercises I do, so when I saw so many people performing muscle-ups with a swinging motion, it unnerved me. I realize now that the swing is essential to the technique.
Imagine an egg tied to a string. If you held the egg in your right hand and the string in your left hand, when you release the egg, it will swing to the left, hang airborne for a second and it will swing to the right. When the egg gets it's second hang time, this is where you would explosively pull yourself upwards for the muscle-up.
Essentially you are utilizing your body's momentum to send yourself atop the bar. It's important to conceptualize the pivot point you will be rotating with. The navel is the pivot of the human body. A human who jumps off a cliff will rotate around their navel. Imagine if the human had anvils for hands, they would rotate around the anvils as the weight imbalance would shift the pivot point from the navel to the anvils.
Likewise, with the bar because your arms are locked out, the bar is your pivot point. The idea of a muscle-up is to revolve around the bar so that you're atop the bar. If you attempt a muscle-up without a swing, you would pull yourself into the bar opposed to atop it. When you do a muscle-up, you are throwing your body in a half circle, you start below the bar, you finish above the bar.
