Why is any game awesome? Typically because it's fun to play, it has the options to support one or more styles of play, or it's rules are particularly well written, elegant, or hits the right spot in terms of what it covers. To me, Fate has all of those things. I haven't played Fate Core yet (working on that), but I have played Fate-based game and they were great.
It's fun to play because what the people around the table want out of the game inform the rules, and not the other way around. Aspects aren't just indicators of what the players want in the game - they're instructions for how to do it. When a character has the aspect My Favorite Dish Is Revenge, it's saying the player wants to have a game with betrayals and revenge and also communicates the theme and tone for that happens. It's a little capsule of how and why.
From a system perspective, for me it feels wide open. Rather than having to think outside the box to recreate some element in the game, Fate Core encourages redefining the box. It's not so much that Fate Core is the first to introduce concepts such as the treating anything like a character - in other games a car, for example, might have hit points or stats like how fast or maneuverable it is. The difference is that in Fate Core if I want a haunted house, I just say it's a Creepy Old House. I don't have to figure out what kind modifier or rule covers "old" or "creepy".
But more than all that, Fate Core has a number of "Oh shit, I never looked at it that way before!" moments. I've always used some measure of "fiction over physics" in games I've run. I've also used some variation of "Yes and/but..." for years without realizing what I was doing. But die rolls measuring the distance from success, and failure possibly being a success at a cost? That's huge. Conceding versus being taken out, and giving the player a choice in how badly they want to push in a conflict instead of just wearing down hit points until they're dead? That's exactly how I want characters being removed from play to work. The entire Fate fractal? Brilliant in its simplicity. It's like the first time that Basic D&D really clicked, after multiple times of reading it. Or the first time I heard Skinny Puppy and realized how much music there was in non-music. Or when I read Neuromancer for the first time. Or first picked up Tribe 8. Fate Core for me is a perspective-changing experience.
The feeling isn't just novelty either - it changes the way I look at every game. Even when I'm not running Fate Core, there's no putting the genie back in the bottle. Fate Core is part of my gaming consciousness now.