Zombies became popular because they don’t have issues. They’re pure horror. So authors and filmmakers can use them as a threat to their protagonists without worrying that the audience will wonder what the zombies are thinking or feeling.
Zombies combine several things people enjoy watching:
Collapse of civilization. Zombie outbreaks are essentially a force of nature. A zombie outbreak distroys civilization at it’s most basic fundamental unit - people. It converts productive people who are necessary for the functioning of civilization into mindless creatures that serve no purpose other than turning other people into more mindless creatures. It’s fun watching that collapse as communication, economic, medical, military and law enforcement systems lose cohesion and shut down.
Post apocalyptic survivalism. Everyone loves watching a group of survivors try and survive as society collapses around their ears.
Morally unambiguous violence. Zombies are the perfect enemy for dispatching in the most gruesome methods imaginable. They are already dead and mindlessly bent on killing, so it makes it ok to do things to them that most people would find repugnant if done to a living human.
The “Dead” movies were around for quite a few years before the current surge of popularity. Although the zombie genre never really died out, I’d say it picked up in the early 2000’s:
2002 28 Days Later, Resident Evil (movie)
2003 The Zombie Survival Guide
2004 Dawn of the Dead (remake), Shaun of the Dead
This (these?). I think the introduction of “fast zombies” in 28 Days Later and other movies really rebooted the genre, and the OCD-level of detail found in the Zombie Survival Guide facilitated the geeked-out zombie survival plans people are so fond of concocting and debating now. Personally I think the whole thing will be played out in the near future, it seems awfully 5-years-ago as far as fads go at this point.