unhonour'd Dead
Dost in these lines their artless tale relate;
If chance, by lonely contemplation led,
Some kindred spirit shall inquire thy fate....

Gray then spends the next few stanzas having that "kindred spirit" ask somebody else about what the dead man (Gray) was like.  In stanza 29, that person tells the "kindred spirit" to simply read the epitaph that is engraved on the tombstone.  

"Approach and read (for thou canst read) the lay,
Grav'd on the stone beneath yon aged thorn."

The following stanza is the start of the epitaph.  The significance of the epitaph is that Gray is writing his own epitaph.  He's reflecting on his own death that obviously hasn't happened yet.  That's a bit morbid in my opinion, but his lament is a fairly uplifting account of his life.  He says that he had a humble birth, but he also says that he worked hard to gain knowledge.

A youth to Fortune and to Fame unknown.
Fair Science frown'd not on his humble birth,

Stanza 31 says that Gray more or less led a generous and "sincere" life, and Heaven rewarded him for that life.  The epitaph ends by telling readers to not worry so much about other details because they don't matter.  They don't matter because Gray is dead.  

This long poem, a series of stanzas treating various natural phenomena at their ends, and set in a tranquil twilight setting (itself the ending of the day), memorializes the unheralded, unmemorialized lives of everyday beautiful things, unsung beauty that “Awaits alike the inevitable hour.” Much of the beauty of natural things, desert flowers or ocean gems, display their beauty to no audience, no receiver of their beauty.  The same is true of the persons lying in a country graveyard, not famous or celebrated, but nonetheless beautiful in their simple rural setting; “Far from the madding crowd’s ignoble strife,/Their sober wishes never learned to stray”.  The epitaph that ends the poem, putatively written on a simple gravestone and addressed to some unknown country person who lived his life, “A youth to Fortune and to Fame unknown” is really the Romantic poet’s oath: to be the audience, the receiver of nature’s beauty, and to give an epitaph to all the otherwise unacknowledged beauty, in the poems themselves.   

"The struggling pangs of conscious truth to hide, >To quench the blushes of ingenuous shame, Or heap the shrine of Luxury and Pride With incense kindled at the Muse's flame." Gray is recounting the unexpressed and un-acted upon thoughts and emotions of average, everyday people, all those who never got their names in history books or anthologies of poetry, the quiet masses, and his main point here in this stanza is that they had opinions and they struggled with their consciences and anguished over staying quiet about their thoughts. They struggled not to voice their opinions but to keep quiet about them; they wrote no poems (“heap the shrine of Luxury and Pride”), and did not rise up in opposition to the injustices, even though they saw them, just as famous rebels and fighters (Cromwell and Milton are mentioned in stanza sixteen). “Ingenuous” means “showing innocence.” Gray is praising all mankind for having ethical and poetic thoughts; even though they were never recognized during their lifetime, they were just as noble as those we honor publicly. This graveyard is filled with their spirit, and Gray is finally giving them the elegy they all deserve; for him they are like "flowers" that did not "waste their sweetness on the desert air", because Gray acknowledges them here.

I assume you mean Thomas Grey's "Elegy Written in a Country Churchyard" (1751). Grey's " Elegy" is a lovely poem, rich in description and meaning. It isn't a formal elegy in any sense (since it doesn't really eulogize any particular individual) but is more of a reflection on death and the fragility of life.

The poem is set in a small country graveyard, and the poet is moved by this setting to reflect on death—and by extension, life. He praises the lives of the simple working people buried in this country churchyard.

Let not Ambition mock their useful toil,Their homely joys, and destiny obscure;Nor Grandeur hear with a disdainful smileThe short and simple annals of the poor.

He suggests it is perhaps better to live an unadorned life free from ambition than to seek fame and power at the expense of others. In this sense the poem venerates a certain brand of stoicism.

One of the special features of the poem is the language, which captures some of the stylistic informality of an earlier linguistic age. In this respect it sits on the cusp of the transformation from Shakespearean English to the modern, rigid form of the language we know today.

Finally, the last few stanzas, "the epitaph", are almost an attempt on the part of the poet to conceive of his own elegy:

Here rests his head upon the lap of EarthA youth to Fortune and to Fame unknown.Fair Science frown'd not on his humble birth,And Melancholy mark'd him for her own.

Thomas Gray's "Elegy Written in a Country Churchyard" shifts, or turns, after line 88.

Up until that point, the speaker focuses on the dead in the churchyard, or cemetery.  The dead in the churchyard were poor and uneducated, and, therefore, their potential was unfulfilled.  The speaker is remembering them and contemplating them. 

But in line 89 the speaker shifts the focus to himself.  His line of thought at this point is that maybe someone will remember him once he is gone.  Perhaps, "some hoary-headed swain [white-haired country laborer]" may notice that he doesn't come around the churchyard anymore; will notice that he is gone. 

The stanza that marks the turn follows:

For thee [himself], who, mindful of the unhonored dead,
Dost in these lines their artless tale relate;
If chance, by lonely contemplation led,
Some kindred spirit shall enquire thy fate [ask why he doesn't come around anymore],... (lines89-92)

Being remembered after death has been a central focus and theme of literature at least since the ancient Greeks sought to perform heroic deeds so they would not be forgotten.  Here, the speaker hopes that someone will remember him, as he has remembered the "unhonored dead."