Muscles require energy to maintain and repair. Your heart and lungs are required to transport and create this energy in the first place.
If your muscles are demanding energy to repair and maintain themselves, this will help you burn fat.
If your heart is in great shape and your body is functioning properly to convert food and oxygen into usable energy, this will help your body burn fat.
If your muscles are demanding additional energy to recover, and your heart and lungs are working harder and providing more of this blood and energy to the recovering muscles, then your body is working at max efficiency in all respects, which means faster muscle growth, faster recovery from injury, and more calories being burned to maintain this improved level of function.
If you're trying to grow muscles in an environment which can only provide so much energy and oxygen, you're only going to get a certain level of results for your effort. If the systems your body uses to recover these muscles are working more efficiently, your muscles will recover more efficiently, and you'll get better results all around.
So, make sure your heart and lungs are in good shape, then make sure EVERY muscle you can manage to work is tired. Large muscle groups are good to work because you can make your body recover over a large surface area without doing a huge variety of workouts to target the same surface area over several smaller muscles. However working your smaller muscles will give you the exact same advantage, you'll just have to be thorough in your exercise to make sure you work them all. With a weak cardio system, this whole recovery concept will only work at a fraction of the speed and efficiency.
There's no tricks or keys. Anything 1 muscle does will impact the entire system of your body in some way. The only trick is to maximize everything to develop a synergy of fitness between a collection of healthy and well functioning systems. Otherwise you really are just pushing yourself to achieve the minimal requirements.
Furthermore, I think it's important for your own health and safety to consider this. There's a strong tendency for 'fit' people, men particularly, to focus a lot on their strength training over their cardio workouts. We also see a lot of fit people collapsing from strokes, heart attacks, and other medical problems you wouldn't expect to see in someone eating well and exercising. I'm not going to pretend this is carved in stone as fact, or taught by training experts, it's just personal opinion. But when you're training your muscles to a fitness level beyond that of your heart and lungs, you're basically doing exercises that those organs would never be expected to perform. Since lifting weights isn't based off your heart, you can complete the exercise. But rest assured you're delivering a harder, and faster shock to your system than that system would normally expect given the level of fitness its been adapted to. People didn't thrive by standing around lifting weights. We thrived by hunting and gathering, which would have combined both strength AND cardio. Our bodies would not do much growth in one respect while getting little development in the other. For this reason I thinks it's important for long term health to maintain an even level of fitness throughout all your body's systems. Best case and point example would be in someone starting out running with weak legs. Sure, their heart and lungs will allow them to keep running, but their legs just hurt too much to continue and they might even get injuries in their legs. I personally feel that this works both ways for cardio and strength training, and thus, my advice is to keep a balance of maximized effort throughout the entire system and try not to develop anything excessively further than anything else.
The exercises we're doing do not reflect the activities our bodies were designed to perform. We have a lot more freedom to control how we develop, and a lot more freedom to train ourselves in ways, and focus on aspects of our fitness that we'd never have the luxury of isolating and giving so much extra focus towards. It's my personal, and humble opinion that because of our freedom, our concepts of beauty, and ideals of self image, that people often end up training in ways that will produce aesthetically noticeable results, and often neglect the systems of our body which show very few aesthetic improvements. It's my opinion that this creates a state inside your body that parts of your body are not properly equipped to meet the demands for, and over time, these systems get stressed.
