Note also that, as implied in Poly’s and EM’s posts, it wasn’t the Hebrews themselves who originally thought up this sort of system. Earlier Semitic languages were also represented by alphabetic scripts derived from the Proto-Canaanite alphabet.
While this kind of abjad or incomplete-alphabetic system that explicitly represents consonants but not vowels may seem like an illogical invention, it had some definite advantages over earlier script forms. Cuneiform scripts, for example, were syllabic, with one (or more) glyphs for each different syllable. So writing similar syllables like “ga”, “gu”, “gi”, etc. wasn’t just a matter of changing one vowel-letter, it required a whole different sign for each separate syllable. Memorizing just twenty-odd symbols for writing consonants would be a lot easier than memorizing a few hundred!
(Not suggesting, btw, that the earliest Semitic alphabets were deliberately devised to avoid the cumbersomeness of syllabic scripts. Syllabic scripts such a cuneiform might not even have been known to the early alphabet-users, who seem to have been influenced by Egyptian consonantal glyphs.)