Well, I'm glad you asked, Fletch.
Your muscle undergoes minor tears when you exercise more than you are accustomed and this damage is reported to you by soreness. This damage is repaired in a couple of ways that build stronger and/or more efficient muscle.
The repair process reduces 'inhibitory' feedback - the feeling that you can't do any more that your muscles produce when they tell you might get damaged - allowing you to do more and get stronger by increments without damaging yourself.
Exercise improves bone density too, making it stronger. Your joints are made up of lots of bone and muscle, so exercising them makes them stronger in that respect and keeps them and the tissue around them healthy and regenerating well. Joint ache from exercising is often just muscle ache in your joints (obvs not if you have an underlying injury or medical condition).
Small amounts of inflammation from damaged tissue can cause stiffness, but the increased blood flow from the inflammation mends you - I don't know if or how this makes the tissue stronger.
Over-exercise can fuck you up, of course. Especially cartilage, where inflammation releases enzymes that break it down, but the right amount of the right exercise keeps it healthy.
Your lungs burn because you require more oxygen than your lung capacity can comfortably supply, making you breathe hard and dry out your trachia and bronchial tubes. It does not reportedly damage you, and exercise expands lung capacity, though I don't know how.
The heart pumps more when you are exercising. But, this makes it elastic, stronger - as with training any muscle - and have a larger capacity. When you're not exercising, therefore, it is pumping less, and more efficiently. Over-exercise can be detrimental to the heart, though.
