I've been engaging in amateur game design most of my life. I think I was 10 or 11 when my friend and I first attempted to design a game system, inspired by Groo: The Wanderer comics and an annual of the Alf comic that featured a mini parody RPG called Luncheons & Flagons.
When Marvel started putting stats on the back of its Superhero trading card sets in the early 90s (starting with Marvel Universe series 2), my friends and I designed a rather simple game system around the stats. It wasn't great, but i when the most recent Marvel Universe RPG tried to do the same thing, I happily concluded that mine was at least usable.
Like most of my RPGs, it was never completed. In Junior high some friends and I shared the game design bug and started writing games based on the Duck Tales & Darkwing Duck universes. We even got it in our heads to do a RPG based on the film/board game Cluedo.
These days I actually try to complete my systems, such as the one I'm currently working on which is about supernatural occult detectives in the vein of Hellboy, The Goon, Mr. Monster, Carnacki, etc. I've been playtesting it using an adaption of the Masks of Nyarlothotep, and I've got a fair bit of the system typed up, as well as some personal deadlines.
As for motivation...well, one is simply that it's compulsory. Designing rpgs is as much of a hobby in and of itself as playing RPGs for myself. But then, I also get off on creating conlangs and re-translating medieval epics, so make of that what you will.
Secondly, as a GM, I know exactly how I like to run games and how mechanics aid or hinder my personal GMing style. So, my games are crafted to specifically make my job as a GM as easy as possible while still hopefully offering what for me and my game groups translates as "depth".
Additionally, there's lots of genres that have not been explored (or not satisfactorily used) in published RPGs. And I am one of those "system matters" people...I like having a system that is specifically crafted to emulate a particular setting/genre/or IP license.
So, say I want to play a game based around Grant Morrison's The Invisibles (true story)...well, I could use any number of games to accomplish this, but if I want it to "feel" exactly right to me, I more often than not will resort to a homebrew (sometimes a mish-mash of other RPG systems, sometimes something completely new.)
And, there have been those odd times where an idea for a dice mechanic comes to me that I haven't yet run across in an RPG, and I will try to build a system around it. Again, just as a creative excercise, though it has left me with notebooks full of unique bare-bones system resolution mechanics. Sometimes they find a second life if I come across a setting that I think fits.