Cursive forms of Hebrew arose in every Jewish community during the Second Exile to communicate or write in Hebrew in secular contexts (i.e., when not writing Scripture). As in the case of the cursive forms of hieroglyphics in ancient Egypt, and the lowercase letters and italics in the Roman script in the Middle Ages, the cursive forms of Hebrew developed in a bid to write the printed letters more quickly. Thus, the three-strokes of printed letters such as aleph, bet and tzadi became just two-stroke ones; and two-stroke letters such as gimmel, dalet, etc. became continuous, fluid strokes, etc.:
Evolution of cursive aleph and dalet (from “Aleph Through the Looking Glass”)
Interestingly, each community took its graphic cues from its Gentile surroundings. Thus, the cursive Hebrew of Ashkenazi Jews in Germany and Poland looked a lot like the Gothic of their neighbours:
Ashkenazi semi-cursive, 1406 CE (from: Beit-Arié, Hebrew Codicology and Paleography)
and the writing of Sephardi Jews of Spain and North Africa was clearly inspired by the Arabic:
Late medieval Sephardi Jewish cursive, ca. 1330 CE (MS. 496, Bodleian Library)
The modern cursive Hebrew is based on the cursive of Ashkenazi Jews of the 18th century onwards.
(Originally written in reply to a question at Quora.com).