"a penny saved is a penny earned", a very old proverb. In fact the proverb here is so out of place that this discussion just lost its charm. But the saying says a lot. When read as "a calorie saved is a calorie earned", it makes sense, at least in this context. So getting back to how hobbies work, Hobbies are those activities that we choose to do, since we love or cherish doing them. Or better, passionate about doing those activities. Be it by nature or nurture, we all develop our unique interests and there is a part or module of our brains that is very capable of handling certain activities better than other people can.
Some paint, some sing, some write, some dance and some solve sudoku and crosswords. Not all can do everything. We do what we do best. But our jobs don't necessarily have to be the kind of activity that we might relish doing. You might be a painter working as a Sales person. Now this is tough, at least until when your brain learns new ways to efficiently form neuronal patterns that can handle new activities better, like remembering names (memory), mailing, thinking, decision making and calling, all at the same time (multi tasking) etc. Most of which are something not generally the kind of things that introverts do. So in the process of doing these new stuffs, our brains consume huge amounts of energy, simply put, more glucose and oxygen. So after a while (until the stamina gradually increases with practice) we are bound to feel tired. That's just normal and fair. For the brain needs rest now and then, to re-energize and form better patterns. In order to make something out of the gathered information, we humans need rest.
Hobbies, on the other hand, are activities our brains are better capable at handling. The patterns of neurons that makes a painter a painter, is unique to him/her. Your unique thinking patterns process data more efficiently and effortlessly. Passion fuels these activities. This makes the brain exercise a different part of the brain, which is already well designed to handle that specific and unique favorite hobby or hobbies of yours. So, you spend lesser energy as you indulge in your hobby, and by this time, the tougher job's data has been sorted and you are good to go again.
Not to mention that you are emotionally charged again and the hobby facilitated dopamine and endorphins are doing their magic. Congratulations, you just got more productive.