Heathcliff withdraws from the world and eats just one meal a day.
Note the similarity to Catherine's own fast in chapter 11.
A few nights later, he spends the entire night walking outside. When he returns to Wuthering Heights, Cathy remarks that he is actually acting pleasantly. He tells Nelly that "Last night I was on the threshold of hell. Today, I am within sight of my heaven."
Note the similarity to Catherine's own night outside after Heathcliff left. Heathcliff's heaven, simply, is to be with Catherine.
Heathcliff refuses all food and demands that he be left entirely alone. The next morning, at breakfast, Heathcliff terrifies Nelly when he seems to see an apparition. She can see nothing, but it seems to her that Heathcliff is communicating with it. That night, Heathcliff again seems to be speaking with a ghost—Nelly hears him say "Catherine." When Nelly speaks with Heathcliff, he reminds her of his burial wishes.
Now Heathcliff's refusal to eat is identical to Catherine's. As his actions mirror Catherine's, he begins to actually see and speak with her. He is being haunted by her, just as he always wished.
The next day Heathcliff locks himself into his room and refuses to even see the doctor. The next morning, Nelly uses another key to get into the room and finds Heathcliff dead and soaking wet—he had thrown open the window to let the rain come down on him.
In his final act Heathcliff opens the window, to connect with nature and with Catherine.
Heathcliff is buried as he wanted, next to Catherine, while Cathy and Hareton are soon to be married and will move to Thrushcross Grange.
Heathcliff's death lets him be with Catherine, and allows Cathy and Hareton to also be together.
Lockwood leaves Wuthering Heights and walks through the moors to the churchyard where Heathcliff, Catherine, and Edgar are buried. He writes that though the local villagers say that they have seen Heathcliff's ghost and another spirit walking the moors together, he "wondered how any one could ever imagine unquiet slumbers for the sleepers in that quiet earth."
Lockwood now sees nature as "quiet," since nothing now stands in the way of Heathcliff or Catherine, who were themselves forces of nature in their incredible and all-consuming love for each other, from being together.