Rob Rendell explains the breakthrough that made stunts "click" for him, in response to a general question on the mailing list about Stunts and Aspects.
Since you mention stunts, I assume you're playing Spirit of the Century?
Aspects are explained extensively in the SotC rules in their own chapter (starting on page 33). I'm pretty sure I can't do a better job of explaining them than the text of that chapter.
Stunts took me a while to come around to really understanding. Superficially, they grant all sorts of random "always-on" abilities that your character enjoys, and are detailed in their own chapter in the SotC rules starting on page 115.
When I first read SotC, stunts felt at odds to the Fate 2.0 way of doing things to me. Since then, however, various things I've read have finally gotten the idea through my thick skull of what Stunts are: they're cool things that anyone with appropriate aspects could reasonably do by spending a Fate Point, locked down to allow whoever has that stunt to do that cool thing over and over as a shtick *without* having to spend a Fate Point. You're sacrificing versatility for repeatability.
The example that really drove the point home for me was the racial aspects in Spirit of Swords and Sorcery: the only mechanical effect of being a Dragon is a single racial aspect, "Dragon". To do dragon-like things, you can spend a Fate Point to invoke the aspect to fly, or breath fire, or shape-shift into human form, or various other dragon-y things. This worked for me as a racial aspect: it was *possible* for you to do these cool things that a regular human character couldn't, but the requirement to spend a Fate Point each time you wanted to do one of those things meant that it was limited in its frequency.
And then I got to the stunts for Dragons, and there was a stunt that mean that you could fly without spending a FP, one which allowed you to breathe fire without spending a FP, one for shape-shifting with the FP. Suddenly, I grokked what all the stunts in SotC were about: as I said above, cool things that your character probably *could* do by spending a FP, but because you want it to be one of the defining shticks of your character, you spend one of your stunts to be able to do it over and over without the FP.
Hope that helped.
 Things that I read which helped me come to my understanding of stunts: discussions on this list, reading Rob Donoghue's "Going Stuntless" article and the conversion "Spirit of Swords and Sorcery" (both of which are in the Files section of the yahoo group) and discussions of how buying Stunts reduces your refresh (the number of Fate Points you start the session with) in the Dresden Files RPG.