As we play we gain experience. The players know when they can level up because they can either see if they have enough experience or the DM will tell them they can level up. How does the character know?
Example (Wizard): a first level wizard has two spell slots and six spells. When you reach level two you have three spell slots and know two more spells.
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When would the wizard know that she'd leveled up?
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Would it only be in an instance when she needs to cast third spell
and finds she can? This would tell her it's time to learn to more spells.
I know that players know, but I was wondering how the characters know?
Best Answer
Character levels are an abstraction of continuous growth
The characters themselves know that they are getting stronger, but they don't experience discrete jumps in power within the game. Instead, they steadily get stronger as they get more experience. The players and the DM use levels to represent that advancement in a useful way.
To use your wizard example, the PHB (pg. 114) states:
This means that in-universe, wizards are constantly gaining insights and conducting research, and thus continually developing and finding new spells as they adventure. The (in-universe) wizard knows that they have new spells when she researches or finds a new one, regardless of level.
However, it's difficult to represent this continuous, somewhat random improvement in a tabletop RPG, which is why the (5e) rules have discrete levels. Just as the initiative order is an orderly abstraction of a chaotic battle where everyone's trying to act at the same time, so too is the leveling system an abstraction of steady progress.