I first wanted to learn to solve it, like Sarah says, just to see a solved cube in my hands. After I knew the beginner's method well enough that I didn't need a cheat sheet, and could do it from memory, I was enthralled with my newfound cube solving superpower for about a month. At the end of the first month I was really starting to get bored with it to be honest. At that point I discovered the Fridrich method, and the promised 17 second averages that were possible (I was at probably 1-2 minutes on average at this point). I honestly don't really remember very well my decision to learn the Fridrich method. I remember looking into Petrus at the time and I remember being confused by it as well (I regret now not learning it then). Once I discovered Fridrich I think the combination of being bored with the beginner method, and the possibility of being really good at something (17 seconds seemed impossibly fast, but it also piqued my interest) made it an interesting challenge that I wanted to give myself. Get to 17 seconds - ... just because it's possible.
I think my true addiction to cubing started the first time I ever saw a solved cube. This was before I even learned to solve it, since I had to disassemble my cube and fix it being in an unsolvable state (as described by the solution guide I was using). I think part of the reason I chose to stick with speedsolving is that I knew I was getting bored with the beginner method, and yet I was still fascinated with the cube in general. In order to stick with cubing I had to just take the next step and learn new stuff. It's been a snowball effect ever since.
I'm curious about how your views of your cubing/speedcubing has changed from the time you first did solves up until now. For those who are still trying to get faster, do you have a specific limit in mind, or are you waiting to figure it out? What will happen when you reach this point? For those who believe you've gotten as fast as you can, what do you think about finally getting to it? Is your current average something you expected from the very beginning?
Just on a personal level I feel that my attitudes have changed quite a lot over the years. First I was in the excited to be learning Fridrich phase. Then came the OMG I know
ALL of Fridrich phase! Then the excited to be improving steadily phase. Then came my first major plateau. Still stuck in the first major plateau. Still stuck.... still stuck.... Stuck some more... FINALLY break this plateau and improve quickly to sub-20. Lots of shorter lived plateaus later until I finally hit my current plateau of 15 seconds global average with the occasional sub-14 average on a super-mega awesome day. This gets me to about 2005. At this point I started to realize that I really have peaked with my speedsolving abilities. My wrist problems will not let me seriously pursue big cube speedsolving, and it limits my 3x3 a bit as well. At this point I moved over to blindfolded cubing. The next 6 years were the excited to finally be improving at something again phase, leading up to today where I am still improving at BLD.
I don't see myself quitting cubing, but I wouldn't be surprised if my cubing interests and focus continue to change. For now BLD is my focus, but in 10 years or so it might be Fewest Moves or maybe even speedsolving again, who knows.