Let's start with the obvious, just to get it out of the way. At one point in Apocalypse, Colonel William Stryker comes to the X-Mansion and captures select mutants who he believes know Charles Xavier's location, and takes them to Alkali Lake. Jean Grey, Cyclops, and Nightcrawler teleport aboard
the transport helicopter and sneak into Stryker's testing facility, where they meet a feral Wolverine. When discovered, Jean sets the beloved antihero loose, who proceeds to mercilessly kill all of Stryker's men, leaving behind quite a mess. This sequence sets the stage for the film's post-credits scene, and (presumably) an upcoming villain.
In that post-credits scene, a paramilitary group is shown cleaning up the mess Wolverine left behind, while men dressed in suits head straight to where Wolverine was being kept captive. They take a vial of his blood, as well as a number of other things, and put it in a briefcase. When that briefcase closes, the Essex Corp logo is revealed; a not-so-subtle reference to Nathaniel Essex aka Mr. Sinister, one of the most iconic and long-running X-Men villains in history. He is a genius-level polymath who, in addition to having telepathic, shapeshifting, and regenerative powers, clones mutants for the purpose of perfecting human evolution.
While Essex Corp is taking vials of Wolverine's blood, he's not the only character Mr. Sinister is a threat to; Mr. Sinister is the perfect villain to affect the stories of several mutants going forward, especially Scott Summers (aka Cyclops) and Jean Grey.