doesn't it mean the body is using up the fat, while it uses the nutrients in a new healthier diet to build the muscle?
Yes, pretty much, except that you can't specifically say that the energy used to build muscle is coming from fat. It could just as easily be coming from the food consumed (despite the caloric deficit), with the burned body fat being used for non-resting energy expenditure (i.e. movement). Or this energy could be coming from glycogen stores, which food consumed would then replenish, and the burned body fat could again be used for non-resting energy expenditure. The point is that with a few exceptions (e.g. the brain's inability to burn fat), energy from food, from fat stores, and from glycogen stores are best thought of as being pooled together, rather than individually used for separate forms of expenditure, and hence it doesn't make sense to describe the fat as being turned into muscle.
I would simply describe it as muscle protein synthesis requiring energy, and people with high body fat levels having sufficient excess energy that a reduction in food intake would not starve the body and interfere with functions like muscle protein synthesis in the same way that a reduction in food intake would in a person with very low body fat levels.
You could also consider why simultaneous muscle gain and fat loss is rare by considering the factors that lead to each of them:
Factors that make you more likely to gain muscle:
- Resistance training
- Protein consumption
- Energy availability
- Not already having a lot of muscle
- Getting plenty of sleep
- Anabolic hormone levels
Factors that make you more likely to lose fat:
- Consuming a caloric deficit
- Resistance training
- Having plenty of fat to lose
- Getting plenty of sleep
Here, the difficulty in fulfilling both of these lists is that consuming a caloric deficit reduces your available energy. But as this happens to a far lesser degree in a person with high body fat levels, and that person will also find it easier to lose fat than a thin person, it's easier for that person to achieve simultaneous muscle gain and fat loss, especially if they tick off all of the non-contradicting factors for each.
In a person who has already gained muscle and has low levels of body fat, a caloric deficit will reduce their energy availability, which combined with their already high levels of muscle will make it near impossible to gain more muscle, unless perhaps they're taking exogenous anabolics.