Edgar Linton's death is chiefly significant in that it marks a change in Heathcliff. When the sexton digs out Edgar's grave next to Catherine's, Heathcliff bribes the sexton to pry the lid off Catherine's casket, so that he can see her. This gives him some peace, as she has not changed so much. He also has the sexton knock out the side of her coffin closest to where he, Heathcliff, will be buried and to promise to do the same to his own coffin when the time comes.

All of this helps Heathcliff cope after his 18 years of agony. He becomes quieter now, waiting for death. Most significantly, he loses his will to destroy the relationship between the young Cathy and Hareton. He has wanted to gain revenge on Edgar and Hindley by destroying the happiness of their children as his happiness with the older Catherine was destroyed. Now he watches them, with some anguish because of how much they remind him of the past, but as he says:

My old enemies have not beaten me; now would be the precise time to revenge myself on their representatives: I could do it; and none could hinder me. But where is the use? I don’t care for striking: I can’t take the trouble to raise my hand!

He says this is not because he suddenly become generous or kind-hearted; he is simply past caring. He says a change is coming, and, in fact, he soon will die. His loss of interest in harming Hareton or Cathy allows the wounds of the past to heal with their marriage.

In brief, what is the significance of Edgar Linton in Wuthering Heights?

Edgar Linton represents civilization; he is the order to Heathcliff's chaos, and reason to Heathcliff's emotional extremes.  Edgar is the "first man in the county" -- a rich squire whose house, significantly down the hill from Wuthering Heights, and considerably newer and more comfortable, is called Thrushcross Grange.  The difference between the two houses - The Heights and The Grange -- is quite marked.  This mirrors the difference between the two men; while Edgar is a gentleman, comfortable in drawing rooms and reading books (and reaping the profits of being a gentleman farmer), Heathcliff is more at home in the wide world, hunting and roaming.

To Cathy, Edgar represents stability in the world, while Heathcliff is all the wild (and attractive, if dangerous) instability in that same world.  She chooses Edgar, possibly for selfish, mercenary reasons (Edgar was richer and would give her better social position than Heathcliff), or, possibly, because she knew that Heathcliff, while intensely attractive, was not good for her.  Edgar also represents legitimate (in those days, mostly inherited) authority.  His parents were known, and he received his wealth and position by his legitmate birth.  Heathcliff was adopted as a street urchin, with no known parents.  While he eventually gains authority and riches in the world, he does so by his wits, and, it is implied, by nefarious means.  He tricks and tortures the Heights out of Hindley,...