The public couldn't discern Swift's prank from any of the other crap that was being peddled at the time, so they ate it up. Of course, Partridge failed to even come down with the sniffles on the day of his predicted death, but that didn't stop Isaac "Star-Crusher" Bickerstaff from releasing a eulogy announcing Partridge's demise, referring to the departed astrologer as "a cobbler, starmonger, and quack."
The fact that it was written in sarcastic verse apparently wasn't a tip-off.
The next day, Swift published a pamphlet detailing all of "Bickerstaff's" predictions that had come true as described, including Partridge's death among them. This would be the same thing as a major news source erroneously reporting the death of some famous celebrity, say Dr. Oz, and, rather than running a retraction, they just keep vehemently insisting that Dr. Oz is dead and eventually include the piece on their Top 10 Stories of the Year list.
And the thing is, since this was several centuries before Twitter or even the telephone, the slow-moving nature of information was what enabled Swift's hoax to gain so much traction. While Partridge scrambled to rush the release of his new almanac to prove that he was totally still alive and still publishing his own equally untrue predictions, everyone else already assumed he was dead, and mourners flooded to his home to inquire about funeral arrangements. This presumably led to much horrified confusion when an exasperated and sleep-deprived Partridge answered the door in his dressing gown. Swift kept the joke going for an entire year before finally revealing himself as Isaac Bickerstaff and admitting he'd made the whole thing up, just to fuck with John Partridge.
"Remember the part where your whole family thought you were dead? Good times."