Hangul was designed, not inherited

Each base consonant is a sketch of the speech organ that makes it. ㄱ is the back of the tongue rising to block the throat; ㄴ is the tongue tip touching the ridge behind the teeth; ㅁ is the mouth; ㅅ is a tooth; ㅇ is the throat. Every other consonant is built from these five by adding strokes, so related sounds look related.

The added strokes are systematic. One stroke more makes a sound stronger: ㄴ becomes ㄷ, and another stroke gives the breathy ㅌ. Doubling a letter, as in ㄲ or ㄸ, gives the tense sounds that English doesn't have. Linguists sometimes call hangul a featural script for this reason: parts of the letters stand for features of the sound (Sampson; see Wikipedia).

The vowels come from just three marks, standing for the three parts of the old cosmology: a dot for heaven, a flat line for the earth, and an upright line for a person. Put the dot (now a short stroke) on one side of a line or the other and you get a family of vowels:

Then letters stack into blocks, one per syllable. 한 is ㅎ + ㅏ + ㄴ, packed into a square so it sits neatly beside Chinese characters, which Korean writers kept using for centuries.

Many scholar-officials opposed the script. In 1444 a group led by Choe Malli petitioned the king, arguing that abandoning Chinese characters would cut Korea off from learning. Hangul survived anyway, in letters, poetry, Buddhist texts and books for women, and was even banned for a time under King Yeonsangun in the early 1500s. It only became the main script of public life in the twentieth century; it was suppressed again late in the Japanese occupation, and newspapers today use almost no Chinese characters.

Four of the original 28 letters have since dropped out, including the heaven dot on its own (ㆍ), leaving the 24 every learner meets. The rest work as Sejong's scholars designed them, which is why a hangul chart fits on one page. Sejong's scholars boasted that a quick learner could master it in a morning, and a slow one in ten days (Hunminjeongeum Haerye (1446)).