I am trying to grasp the narrative feeling of leveling up as a character in D&D 5e.
Suppose at the end of a session a Sorcerer has enough experience points to level up. I am using the Sorcerer class as an example due to the innate nature of their powers.
How does that feel to them and how is learning new spells and abilities from one moment to another explained in a narrative fashion?
I like Theik's answer, though I'd like to propose a slightly different take on things.
You could view leveling up as those Eureka! moments that often occur with learning new things. They're the moments when something you've previously struggled with suddenly makes sense, when the movement you've practiced hundreds or thousands of times suddenly happens perfectly without you consciously thinking about it.
If you've ever done any software or web development, you'll be struggling with a problem or bug, then suddenly the answer will come to you and you'll understand the code better.
In the gym, when you've been struggling to put together various ques for a deadlift, rep after rep, chest up, back tight, hips down, sit back (but not too far), bend the bar, spread the ground, then you'll take a breath, empty your mind and pull a perfect rep.
In climbing when you try and same route over and over again, then suddenly something clicks and you understand you've had your body slightly out of alignment, your center of gravity off to one side, and that hold you couldn't grip becomes a lot more positive.
That moment in learning a language when you suddenly realise that you don't have to translate every individual word in your head to understand what's being said.
In movies, like The Matrix, it's the moment that Neo understands the matrix and sees it as it really is in it's pure form. Those moments of clarity when something suddenly makes sense, when during a fight scene the hero's opponent suddenly seems to be moving in slow motion.
In a fight in D&D, when the sorcerer suddenly realises that the spell he just cast at that goblin was instinctive, he didn't have to think about it, meaning he can concentrate on a secondary spell, secure in the knowledge that the first one is well learned.
"Levels" in every version of D&D are a metagame concept, they're an abstraction of the characters abilities, strengths and flaws. Your Sorcerer does not wake up one day with a new level and is suddenly capable of casting a new spell, it's something that has been developing over time.
The thing is, the game can't exactly model this 'over time' process. A wizard is constantly testing new spells and eventually for the metagame itself, they have enough experience to level up and they get new spells. And to the player, it's exactly as if one minute the wizard had x spells and the next minute he has x+2, but as far as the Wizard is concerned, it's something they've been developing over the past X months and is unrelated to that 90th goblin they killed just now.
The problem with metagame information
D&D in all its iterations tries to help you tell epic tales like Lord of the Rings, but it requires some metagame properties to determine what you can and can't do. It'd be weird if halfway through Lord of the Rings, Frodo suddenly started throwing fireballs around, because it doesn't fit the narrative at all.
D&D tries to reinforce that narrative theming with metagame properties like AC, levels, HP, 6 second combat rounds, etc. It decides that Frodo is a level 3 Rogue (or whatever) and that he can't, in fact, cast Fireball.
But because characters develop over time and players like being able to progress, there needs to be an ability to get new stuff to work with over time. This is abstracted, out of character, as levels, but these levels have no meaningful relation with the actual narrative of the game.
And sadly enough, there's simply no way in these games to properly model the whole "I'm working on improving this thing, but I'm not quite there yet", and that's why at some point, you just "level up" and suddenly develop new abilities, which can feel extremely jarring.
Why did the Fighter suddenly learn to cast spells at level 3 in D&D 5E? Because his player opted to pick a class, something entirely outside of the narrative. Inside the narrative, that's something the fighter has likely been working on for ages.
The same holds true for, for example, multiclassing in 3.5. It was entirely possible to suddenly get entirely new and unrelated abilities by picking a new prestige class. That doesn't mean that in the narrative the character just woke up one day with a new ability, it means that the game somehow needs to model these things in a way we can understand.
Where you see a level 6 Sorcerer, the character inside the narrative might call themselves a hedge-wizard from the school of Hocuspocus. That has no overlap with the metagame information, but that's because that metagame information has no in-universe meaning.
Experience points are an abstraction of the experience a character gains by adventuring. By becoming experienced enough characters gain some kind of increased capability. In real world terms, it feels the same as mastering any skill or going to the gym. You get incrementally better, and over time you can do new things. Except there is magic in D&D, unlike real life. This is how we play it at my tables, and it works great and is easily understandable.
Sorcerer's power comes from their bloodline or raw magic. To them leveling up may mean unlocking part of their magical potential, better using their abilities, and becoming more attuned and comfortable with wielding their power.
When leveling up the sorcerer's general abilities increase; their spell slots, hit die, proficiency, and sorcery points. They also gain new spells and abilities, unlocked from experimenting with their powers, new capabilities as a result of training, or clever manipulation of existing abilities.
You may want to break down each separate class feature if you are looking for specific explanations.
