Homebrew is essentially the creation or "brewing" of software or games at home or "in your own garage" (i.e. not by a studio or a business). It is usually a process that is done as a side thing, a hobby and sometimes in the hopes of earning some money but it is rarely, if ever, a serious venture to make a living. Once that happens, it is no longer homebrewing and becomes a job or an indie movement.
Why make homebrew at all? It's quite simple: because I/we can.
Until you become a Developer yourself, you're unlikely to completely grasp the passion and love we have for making something out of nothing on a device we're so interested in, it's as if it were a rock we can find an infinite amount of uses for. To us, the device is indefinitely transforming and improving - that is, until we get bored of it. Where someone sees an electronic device like a PSP or someone sees a rock, we see the potential and many things we can make the device or rock do to achieve something it wasn't specifically made for.
It's not about ego. Not at all. In fact, I resent that remark. It's really simple - at least for me: I'm a Developer because I love expanding the horizons of anything I can get my hands on or I enjoy bypassing any restrictions people put on me/the manufacturer puts on the device.
I do not develop for you or anyone else.
I do not develop for recognition.
I do not develop to fuel my ego.
I do not develop for money.
I have accomplished so much with many different devices, including the PSP, but I choose not to share it. I was considering sharing my work with the PSP Scene, but the attitude that infested it sickened me. In the end, it doesn't matter because I and many others develop for ourselves and if we decide to release it, it's usually as a "I found this cool/useful, maybe you will". Personally, I don't tend to support such releases, but there is the time when I release something and support it with updates, etc. just because I can spare the time. I haven't ever released anything expecting praise for my work and I'm sure the same applies to many others. Still, I won't lie and say it isn't nice when you go online and find your work spreading like wildfire.
I develop things for myself and my own entertainment. I enjoy developing. It is called a hobby and I strongly disagree with your sentiment that people play music to stroke their ego. Sure, some do, I'm not going to disagree with that. But the majority of people play music simply because they enjoy doing it. It makes them feel alive, free, gives them an escape from reality and so on.
I love reading. Do I do it to prove to myself that I am good at reading? No, of course not. Doing so would be ridiculous and I think I'd need some professional help as a result.
I love singing. Do I do it to prove to myself or anyone else that I'm good at singing? No, I just do it because I enjoy it. I already know I'm good at it, I don't need any validation.
I love writing.... you get the point.
The same could be applied to *any* hobby. So, why then, can't the same be applied to (homebrew) development?
As for whether or not you're a Developer - you aren't - by definition. In fact, you're most likely considered a story writer/a writer by your description of what you do. Besides, I'm sure everyone, including yourself, understood what was meant by the use of the term Developer in this context anyway. In your case, as a story writer, I can see where you are coming from and why you feel this way. Writing is meant to be shared and enjoyed by others. Some writers can produce work simply for themselves but most get their enjoyment from having others reading and experience their creativity not just reading their work themselves.
Still, with all that out of the way, homebrew development is a great way to showcase your talent to large corporations and if you bring innovation and quality to the Scene or the market, you're likely to be noticed and often even approached with interest. I was and I know many homebrew developers in several Scenes or areas who have.
At the end of the day, all the work I've done can still be put to use in a portfolio, so none of my time is wasted if you want to look at it that way.
I don't think "Why make HB at all" is the best question because I feel like the same Developers you are asking the question to would be Developing things regardless of the answer. I think a better question would be: "Why release your work at all?" Now there's a topic I'd like to pick some brains on.