Going through the Fate learning curve coming from more traditional things, there's a few things that really had to sink in. I think I've been pretty successful at helping others get through it, so here's the top things that I've learned.
This is the biggest thing. By fiction I don't mean "the GM's pre-planned story." I mean the stuff we imagine in our brains. In Fate, the imagined world drives the game, not the rules. We use the rules to help us figure out what happens when it isn't immediately obvious from the situation.
About 80% of the questions that I've seen new GMs/players ask about Fate are resolved by this. Start with what's happening in the actual world, and then go from there. Does an aspect go away? Well, when the thing that it represents isn't true or doesn't make sense any more, it goes away.
For players and GMs, one of the things that this means is that you generally don't want to declare your action in terms of the rules. In Core and especially in FAE, you declare what you're doing in terms of the fiction of the world, and then figure out how to use the rules to figure out what happens. This has the side effect that nonsensical actions never make it to the dice rolling part - they get stopped before then as the GM or players or both say "uh, that doesn't make sense, no."
Even aspects follow this pattern - a room may have a Dark aspect because it's dark. The room isn't dark because it has a Dark aspect.
Fiction, not Physics
Another one of the cute Fate taglines that actually makes a lot of sense when you get it. Many games try to model the real world, or even "a" world. We have turns that take some number of seconds, and when we look at how rules work, the easiest way to think about them is how things work in real life.
Not so with Fate. Fate could care less about how things work in the real world. Fate is deliberately designed to work the way that fiction works. It models the way that writers write books, movies, and TV shows.
So, a turn in Fate? How long does it represent? Trying to think of it in terms of an actual length of time is basically incorrect. The best way to think of it is as a camera shot. How long does it take? As long as the camera needs to show the action. Making something an aspect is like Chekhov's gun, a camera shot pointing out the feature of the scene, and alerting the viewer that it will be important.
Even Create Advantage matches the very common fictional rhythm of establishing something, and then paying it off.
If you try to map Fate's mechanics to real life, you'll find yourself scratching your head. If you try to map Fate's mechanics to movies, or TV shows, or books, you'll find that it matches better than almost any system out there. A thing that people start doing once they're really internalized this is, when watching stuff, to start mapping the screen action to various mechanics. "Oh, those big geysers on the planet, those are definitely an aspect. And Star-Lord's ship got hit with one, that was definitely a compel."
The Purpose of Scenes
In many traditional games, an encounter or scene exists as a kind of a test (one the players usually win). "Can you beat this set of bad guys? If so, you get to keep going!"
That's really, really not how Fate does things. In Fate, the reason for just about any kind of scene is to figure out what happens. A scene is a fork in the road of the story, not a gated challenge that must be overcome.
This means that in a good Fate game, things don't go the way players want them to on a regular basis. And that's okay! Usually when I do intro scenarios, I'll even deliberately throw a near-impossible fight at players early on so that they can exercise the Concession mechanics, and see that losing a fight just means that the story progresses in a different way than what they had wanted.
Stress Isn't Damage
This is getting a bit more nitty gritty than the more high level, conceptual stuff above. But it's a common sticking point, and so is worth a mention.
Stress isn't damage. Arguably, Fate doesn't have a damage system (Consequences come close, but although most "damage" is best modeled as a Consequence, not all Consequences are damage). Stress is, mostly, about pacing - "how long does this fight scene last?" In a movie, think of a fight scene and all the back and forth that goes on. A character does a really harsh series of blows to someone, and they're on their heels, but then they turn around and kick them in the stomach. Or some such.
That's all Stress. That's exactly what Stress models. So that big Stress hit? Yeah, don't narrate that as the guy getting ripped over crotch to sternum. It doesn't make sense, because stress goes away at the end of the Conflict. Save the solid hits for taking Consequences, because that's really the best place for them.
Aspects Are True
A common misconception among new Fate players is that aspects only impact things when invoked or compelled.
First off, PC aspects can definitely have an impact beyond that. If you're a Princess of the Realm, you can get into the castle, unless something weird is happening. A typical peasant might not, but you're the freakin' Princess, so you get in.
Situational aspects can also do things beyond invoke/compel. If you're On the Ground, you can't run to the next room. Because you're on the ground. That makes sense. Remember fiction first? Well, the action doesn't even get to the rules because it doesn't make sense with the facts we know.
For things that make things harder but not impossible, the best way to handle them is typically passive opposition. That will increase the difficulty if there's no active opposition, and if there is active opposition, the passive opposition acts as a "floor" on their roll. So if your opponent is On the Ground and is trying to hit you, they might have a passive opposition of +2. Now, no matter how poorly you roll on your defense, the worst you can effectively get is a +2. This can do a good job of modeling the facts of the situation without getting into the situation of chasing down bonuses to add to your roll like you do in many games (which can make rolls that can never fail and other un-fun stuff).
It's a Different Game
Really, this is the big one. Fate isn't D20. It's not D&D 5e, or GURPS, or BRP, or any other system. It's Fate. A huge amount of the problems that I had in learning Fate were not recognizing that, and trying to use my decades of experience and apply it to Fate.
Bad idea. Just read the book, and do what it says. If you find yourself thinking "oh, they really mean this" or "that's like this other thing" or anything of that nature, strike that thought from your mind. If you think "well, in this other game, I'd do this here," banish that thought as well. Treat it like playing a different game entirely, and not just as a different version of your favorite game.