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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.

  • When would the wizard know that she'd leveled up?

  • 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?

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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:

The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse. You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

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.

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    \$\begingroup\$ Yes, exactly. Another example would be a person working out in a gym: they steadily get stronger, and are able to lift heavier weights over time. There might be benchmarks (they can now lift 100 lbs, when they could only lift 50lbs before), but the improvement is still continuous. \$\endgroup\$
    – Icyfire
    Feb 5, 2017 at 17:15
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    \$\begingroup\$ Worth noting that there are systems where progression is modeled much more incrementally, such as the Storyteller system from White Wolf/Onyx Path. In those systems the player continuously buys individual new powers with small bundles of experience points. \$\endgroup\$
    – Grubermensch
    Feb 5, 2017 at 17:47
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    \$\begingroup\$ This is true of nearly every point-buy system. \$\endgroup\$
    – keithcurtis
    Feb 5, 2017 at 18:12
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    \$\begingroup\$ Beyond wizards, you might also think of various martial arts styles. In many of these, there is constant practice as you are taught a new technique, spend some time mastering that new technique (say, a new kind of kick or block), along with other, similar techniques. Then, at some point, your teacher encourages you to take a test to prove your mastery. At that point, you gain a new belt color. The level-up represents the moment you're confident you've mastered the new techniques - you went from yellow belt to green - but you've already been practicing them regularly. \$\endgroup\$
    – CaM
    Feb 6, 2017 at 13:25
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    \$\begingroup\$ The 2e DMG once compared the experience of leveling to waking up in the morning and realizing that you finally understand what a cosine is. The metaphor probably still holds. \$\endgroup\$
    – The Spooniest
    Feb 6, 2017 at 14:49

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