The poem was written in the classical style, but has elements of the Romantic poets who relished in the power of the individual, the beauty of nature, and the supernatural among other things.  Probably the most important factor pointing to the poem's popularity is that it idealizes the comman man, and brings his importance in the universe to an equal status with those who were wealthy and affluent.

The poem invokes the classical idea of memento mori, a Latin phrase which states plainly to all mankind, "Remember that you must die." The speaker considers the fact that in death, there is no difference between great and common people. He goes on to wonder if among the lowly people buried in the churchyard there had been any natural poets or politicians whose talent had simply never been discovered or nurtured. This thought leads him to praise the dead for the honest, simple lives that they lived.

One aspect of the poem that relates to people is that it mourns the death of common, ordinary people, idealizing and elevating them. Readers are able to relate to the subjects of the poem because they are people who worked hard for their families, much like us. The poet says the common people buried in the churchyard should have memorials on their graves just like the rich and famous do because they could have probably been something if given the chance. This speaks to the dreams we all have to make it big or to do something with our lives. The poem speaks simple truths to readers. Samuel Johnson praised it by saying, "The 'Churchyard' abounds with images which find a mirrour in every mind, and with sentiments to which every bosom returns an echo are to me original. . ." The story is universal and timeless. It is a meditation of how death is the final act of a human being, no matter who you are or how much money you have. We see the world through the speaker's eyes who represents all humankind. But the poem doesn't end with terrible depression. The speaker hopes for immortality by making a friend in Heaven and by believing that he will rest in "The bosom of his Father and his God." The beautiful language of the poem also make it a beautiful experience to read it because the words speak to us in its beautiful rhyme, rhythm, and alliteration.

What explains the popularity of "Elegy Written in a Country Churchyard"?

The popularity lies in the poem's egalitarianism and  in its reminder "Memento Mori" (Remember you will die). 

In the end, possessions and fame will mean nothing. "Far from the madding crowd" we are all alike in death.  Death is the great equalizer: 

Can storied urn or animated bust
Back to its mansion call the fleeting breath?
Can Honour's voice provoke the silent dust,
Or Flattery soothe the dull cold ear of Death?

Another factor contributing to the ongoing popularity of this poem is the acknowledgment that some who die have never been given a fair chance in life.  As the speaker thinks about who might lie beneath those "many a mouldering heap" (freshly dug graves) he wonders what these simple peasants might have been had they means and opportunity to accomplish more in their lives:

Perhaps in this neglected spot is laid
Some heart once pregnant with celestial fire;
Hands, that the rod of empire might have sway'd,
Or waked to ecstasy the living lyre:

But Knowledge to their eyes her ample page,
Rich with the spoils of time, did ne'er unroll;
Chill Penury repress'd their noble rage,
And froze the genial current of the soul.