The same way that other alphasyllabaries AKA abugidas (i.e., syllabaries consisting of consonants modified by vowel markers) did: adding vowel markers to an abjad, and if these didn't start off joined to the consonants, joining them to the consonants.

There are two main families of alphasyllabaries: the Ethiopic (from when the name "abugida") and the Brahmic AKA Indic. These probably all share the same original ancestral abjad: Sinaitic AKA Proto-Sinaitic AKA Proto-Caananite. This appears to have developed into Ancient South Arabian, which was used for inscriptions in the Horn of Africa as early as the 9th century BC, which then developed into Old Ge'ez AKA Old Ethiopic, which then developed into Ge'ez and Amharic. Vowels were added in the transition from Old Ge'ez to Ge'ez.

Travelling the other way, Sinaitic AKA Proto-Caananite became Phoenician, which in turn spawned Greek (to the west) and Aramaic (to the east). Aramaic is the probable ancestor of Brahmi script, an ancient Indian script that gave rise to a wide range of Indian scripts, and from them Central Asian and SE Asian scripts. The earliest known Brahmi inscriptions include vowels, which are diacritics modifying the consonant, except for the default vowel, a short "a", in which case the unmodified consonant is used. Whatever the ancestral abjad (which is usually assumed to be Aramaic, but this isn't certain), it appears that vowels were added when it was modified to better suit writing Prakrit (i.e., the language Prakrit). While there is plenty of general visual resemblance between Aramaic and Brahmi, there are enough differences so that the ancestry is uncertain, and other suggestions for the origin are offered: (a) some other Semitic abjad, (b) the Indus Valley script (but the difference is huge), and (c) a completely new invention, possibly inspired by an abjad like Aramaic, but not developed from it. The modern advocacy of the last two often appears to be more driven by nationalist hopes rather than the hope of a genuinely correct answer.

Aramaic, Hebrew, Arabic, and other semitic languages always use abjads, which makes sense because these languages are uniquely suited for it because of their consonantal root system

First, these languages don't always use abjads - both Arabic and Hebrew have optional vowel marker systems. The difference between these scripts with optional vowel markers and alphasyllabaries is that the vowel markers are no longer optional. Also, this "no longer optional" isn't always a strict rule, as there are examples of Brahmic scripts in India being used as abjads, omitting the vowels, into and past the medieval period.

Alphasyllabary scripts can be used for languages where syllables don't always start with consonants by using a silent consonant or "zero consonant". One sort-of example of this is Korean hangul, probably inspired by Mongol Phagspa (not strictly an alphasyllabary, because the vowels are written separately), in turn derived from Tibetan 'Phags-pa (which is a true alphasyllabary), where every syllable block must start with a consonant, and the silent consonant is used where necessary.

Finally, there is a recent family of alphasyllabaries, Canadian syllabic writing, developed for writing Cree in the mid or early 20th century, inspired by the recently-invented Cherokee syllabary, and drawing on Devanagari (an Indian alphasyllabary) and Pitman shorthand. The vowel is indicated by rotating the consonant. Indicating vowels by modifying the initial consonant came from Devanagari (and is what makes it an alphasyllabary) and using rotation came from Pitman. Final consonants came from Pitman.

• A grave marker with Cree syllabics: https://commons.wikimedia.org/wiki/File:Moses_roberts_syllabics.JPG