Well, sort of. Technically muscles burn calories, but it is complicated...
To be honest, I see fitness enthusiasts and Beachbody Coaches using this phrase all the time. Note, there is a vast difference between a fitness enthusiast and a fitness professional, so be careful what knowledge you are gaining from every individual.
Anyways, saying "muscle burns fat" is like saying "my car burns oxygen". Both are correct statements, but they can be misleading.
Body fat, also known as adipose tissue serves many functions in the human body. Insulation and energy are its primary functions. To be honest, the body doesn't really like, or want to carry around more weight than it has to, but because our bodies have evolved to also not want to waste or lose too much energy, you end up storing a lot of your extra calories away when you over eat.
To be honest, muscles' energy 'currency' of choice is something called glycogen. Glycogen is a readily-available energy source stored in the muscle cells as well as the liver. When you perform high-intensity muscle contractions, such as resistance training, your muscles are using glycogen.
Adipose tissue is more of a slow-burning fuel source because oxygen has to be present during the metabolic process in order for it to be a viable and useful energy source. For example, when you are out for a leisurely walk, your breathing and heart rate are more than capable of keeping up with your cells' oxygen demands, so your primary fuel source is usually adipose.
Having said that, a leisurely walk doesn't really require a whole lot of caloric expenditure, so intense exercise is definitely the way to go if you are trying to maintain a healthy body weight. So how does muscle burn fat exactly?
Well, as I said before, muscle actually burns glycogen in intense circumstances. It is a quick and fast-burning energy source. But, when intense exercise is over, your body goes into a state known as EPOC (excess post-exercise oxygen consumption) . Think of a sprinter like Usain Bolt. During his 9 seconds of sprinting, he, along with his competitors are probably either holding their breath, or breathing at a pretty normal rate. This is because the body is using anaerobic means to fuel its muscles. Once the race is over, however, their bodies go into EPOC mode. During the post-race interviews, that is when you see them huffing and puffing as they try and replenish those used energy stores.
That is when adipose tissue is used as fuel by your muscles.
So what advantages can we learn from all of this?
First of all, the fitter you become, aka, the more efficient your body is at utilizing stored energy reserves, the more effective it will become at using stored adipose as a primary energy source.
Secondly, even though your muscles don't directly burn adipose, intensity is definitely the name of the game. An intense, efficient, and effective resistance workout is the best way to go to maintain a healthy weight, as EPOC, as well as micro trauma in the muscle fibers can cause a spike in metabolism and calorie burn for up to 72 hours following a workout.