I'm by no means an advanced DIYer, but I do tackle what I consider simple things: appliance repair, light electrical work (changing an outlet, etc), light plumbing (replacing a toilet, fixing a leaking faucet or drain pipe, etc), drywall repair, painting, sometimes even reupholstering a chair. I'd say frugality came first, then DIY ability. An older cousin was very handy, and when he visited while I was in high school he would change the oil in his car. That was because he lived in an apartment and didn't have a place to do it, whereas we had more space. I learned from him that just putting your hands on something and looking at it was half the battle towards figuring out how to fix it. So when I had my own car, and realized how expensive an oil change was, I did it myself.
Simply being interested in how things work is the first step to becoming a DIYer, and the next is developing the experience to learn that you can do things by yourself. Finally, it's just determination, practice, and refusal to pay someone else for what you think you can do on your own. And somewhere in there is accumulating lots of tools.
