Credited to Ethiopia. The consonantal script derives from Ancient South Arabian writing, and its place of origin is debated among present-day Eritrea, Ethiopia and Yemen; the oldest known example is at Matara in Eritrea. The vocalized system is tied to the Aksumite kingdom, whose rulers Wazeba and Ezana left the first vocalized letters and texts.
How it began
The Ge'ez script grew out of the Ancient South Arabian script. After the 7th and 6th centuries BCE, variants of that script appeared in rock inscriptions in the Tigray region of northern Ethiopia and in Eritrea. By the first centuries CE an Old Ethiopic consonant alphabet was in use. Vowel marking arrived in the 4th century: a coin of the Aksumite king Wazeba carries a single vocalized letter, and the first fully vocalized texts known are inscriptions of King Ezana of Aksum. Ethiopian and Eritrean church tradition credits the vowel system to scholars led by Frumentius, the missionary said to have converted Ezana.
Before it existed
Before vocalization, Old Ethiopic was an abjad, an alphabet of consonants only, with letter shapes basically identical to the first-order forms of the later script. It was written from right to left, unlike the Ancient South Arabian script, which ran in alternating directions. Encyclopedia.com, quoted by Wikipedia, notes that the direction changed to left to right from the 4th century, when Ethiopia was Christianized.
How it works
Ge'ez has 26 consonant letters. Marks for the vowels u, i, a, e, e-schwa and o were fused onto each consonant in a recognizable but slightly irregular way, so the script is laid out like a syllabary: each sign stands for a consonant plus a vowel. The unmodified letter carries the inherent vowel, and the sixth form also serves for a consonant with no vowel after it. Linguists call this kind of system an abugida. Some scholars have suggested influence from the Brahmic scripts of India, which work the same way and were known along Aksum's trade routes.
How it spread
The script was first used for the Ge'ez language, and Britannica records that the Bible was translated into Ge'ez between the 5th and 7th centuries. It was later adapted to other languages, above all Amharic in Ethiopia and Tigrinya in both Eritrea and Ethiopia. Wikipedia lists its use for Gurage languages and at least 20 other languages of Ethiopia.
Where it stands today
Ge'ez itself survives as the liturgical language of the Ethiopian and Eritrean Orthodox Tewahedo churches, the Ethiopian and Eritrean Catholic churches and the Beta Israel Jewish community. The script remains in daily use for Amharic, Tigrinya and other languages. It is encoded in the Unicode Standard under the name Ethiopic, across several blocks beginning at U+1200.