Here's a good explanation from Richard Salomon (2003), "Writing Systems of the Indo-Aryan Languages". Note the startling suggestion that this arrangement might have been systematized even before the advent of writing!

> In this system, usually referred to as varṇamālā ‘garland of letters’, the characters are divided into vowels (svara) and consonants (vyañjana), and each major set is further divided into subgroups, again on phonetic principles. Thus the vowels are subdivided into simple vowels and diphthongs, each set up as short/long pairs (a-ā-i-ī-u-ū, etc.), while the consonants are classified into the groups vargīya (comprising stops, sparśa, including nasals, anunāsika), antaḥstha (semivowels), and ūṣman (spirants, including ha). Within each consonant subclass, the individual letters are arranged by place and manner of articulation. Thus the largest set, the twenty-five vargīyas, is divided into five sets (varga) of consonants with the same place of articulation, arranged from the back to the front of the mouth: kaṇṭhya ‘velar’, tālavya ‘palatal’, mūrdhanya ‘retroflex’ or ‘cerebral’, dantya ‘dental’ and oṣṭhya ‘labial’. Each varga contains five types of consonants: unvoiced unaspirated, unvoiced aspirated, voiced unaspirated, voiced aspirated and nasal. This system may have originally been developed in a tradition of linguistic analysis before the use of writing, and only subsequently been applied to the written form of language in Brāhmī script. The earliest written specimens of the are found in terracotta plaques of about the second century BC showing a schoolboy’s writing lessons (Salomon 1990:271).

The Korean script, invented much later, is also impressively phonetic. https://en.wikipedia.org/wiki/Hangul#Consonants

Note the "g" here is like that in "gum" and not "gin"