Even today, the vast majority of people who even can handwrite the Latin script use some form of cursive. Same goes for users of the Cyrillic script. Americans are among the only people who have a bizarre phobia of handwriting.

People used cursive because it's vastly superior when writing with a quill. Otherwise you'd have to write slower, and you'd probably leave ink drops all over the paper. It then still stuck in most places because it's simply faster and not very difficult unless you have Parkinson's or dysgraphia.

Also, every person who'd have the privilege of literacy would be taught to write like that. This wasn't a particular 'style' really, this simply was, and in most places still is how you write if you're not a child. Also, keep in mind that while old handwriting may be hard to read for us, it would be very common and therefore familiar to the people who lived back then. We may fail to recognize some forms because they're very different from print letters, but they would be completely natural to them.