Why are stats in Angband written as 18/** instead of 19, 20...?

In Angband (and a lot of derivative, and maybe in its predecessors also), when your stats (e.g. DEX) is below 18, it is written with a single number (1 to 18), and after that, if the stat still increases, it is written with the following format: "18/xx", where xx goes from 10 to (more than) 200.

Also, xx increases by more than 1 when you get a +1 item.

What is the intuition behind this notation?

I can only give you a sort of meta-answer, where this system comes from, as I haven't played Angband.

Angbands attribute system is based on AD&D. In AD&D, strength (and only strength) could be represented with 18/xx numbers. 18 is supposed to be the maximum you can achieve without magic means in AD&D (also by virtue of it being the maximum roll on a 3d6), but Gary Gygax wanted to differentiate warriors more, so when a fighter(or paladin or ranger) reaches 18 strength, they get to roll a percentile die (d100) to get a percentile modifier to their strength, e.g. 18/50, or 18/90, where the latter is better than the former and 18/00 (meaning 18/100) would be the strongest person in the world.

This model provided a lot of problems, for example, the gap between 18 and 19 str is huge and if you ever got something, which boosted your strength, you get massively more powerful and this also allows non-fighter classes to leapfrog over fighters (non-fighters never got percentile, so if another class got a magic item boosting strength by 2 they'd go from 17 to 19, while a fighter would only go from 17 to 18/50).

The system was abandoned in D&D 3rd edition, but it still exists today in the Hackmaster RPG, which adds percentiles for every attributes and at every value (so you can be 12/45 or 14/13).

Angband seems to have embraced the concept that 18 is the maximum and cannot be improved upon and decided to go with percentiles at the top all the way. A +1 item improves it by more than 1 percentile, because a +1 is one full attribute point, while one percentile is not.