The poem’s meaning in the story refers to the fact that no one can stay young and innocent for long. 

When Johnny and Pony are on the run, hanging out, Johnny comments about the beauty of the sunrise.  This reminds Pony of a Robert Frost poem.  He connects the poem to the moment because the poem is about how nothing stays young. 

"The mist was what was pretty," Johnny said. "All gold and silver."

"Uhmmmm," I said, trying to blow a smoke ring.

"Too bad it couldn't stay like that all the time."

"Nothing gold can stay." I was remembering a poem I'd read once. (Ch. 5) 

This poem becomes very significant to Johnny.  When he is dying, he asks to speak to Pony.  He quotes the poem to him, which demonstrates how affected he was by it.  To Johnny, the poem has come to symbolize the innocence of youth. 

I barely heard him. I came closer and leaned over to hear what he was going to "Stay gold, Ponyboy. Stay gold..." The pillow seemed to sink a little, and Johnny died. (Ch. 9) 

Pony is deeply affected by Johnny’s death, of course. He understands what Johnny is telling him.  Even though he is a greaser, and all of his brothers are greasers, this does not mean that he has to follow this lifestyle.  He has a chance to be something else. 

Pony has always been the deep one.  He is good at school and likes to watch movies.  Yet, through spending time with Johnny, Pony learns that his friend has depths he hadn’t realized.  Johnny may not be good at school, but he does care and he is a good thinker.  Johnny was very taken by Gone with the Wind, for example.  Pony has a chance to "stay gold," meaning that he can rely on his education and his intelligence to make something of himself in the world.  

Robert Frost's poem "Nothing Gold Can Stay" is about the death of innocence and the transience of life. Basically, all good things must eventually come to an end. Gold becomes a symbol for the innocence of youth, which is inevitably lost to the pressures of long life experience and the encroaching cynicism that accompanies it. Things that inspire wonder in young people, such as a sunset, become routine to adults, who take everything for granted since they have lived longer.

The young characters in The Outsiders associate the poem with their own innocence and the fear of the adult world. Adolescence is a time of transition when teens start to lose both the illusions of childhood and the benefits of the innocence that came with it. For Ponyboy and Johnny's social circle, adulthood or premature death prevent anyone from remaining gold. Therefore, Johnny's urging Ponyboy to "stay gold" is a plea for him to retain as much of his innocence and sensitivity to beauty as he can, even as he grows older. The world might still be a harsh place, but if Ponyboy can appreciate the good and beautiful things within it, then he can remain gold.

The line "Nothing gold can stay" is from Robert Frost’s poem "Nothing Gold Can Stay," which is all about how nothing lasts forever—in particular, the good things in life. Frost uses the word gold to stand for all good, pure things. Just like the substance gold is heralded as a treasure, things that are called gold are assumed to be good, beautiful things. In saying that “nothing gold can stay," the speaker is saying that these beautiful things are temporary.

Frost also references the Garden of Eden in the poem. According to the Bible, this garden was the paradise where the first human beings lived until they sinned and were banished. Through this reference, Frost applies the concept of good things being temporary to innocence and purity.

The line "nothing gold can say" is thus also reminiscent of how the innocence of childhood cannot last forever. This is relevant to The Outsiders because the boys are at an age where childhood innocence is lost and people become consumed with the harsh realities of adulthood. In encouraging Ponyboy to "stay gold," Johnny is telling him to try and keep his innate innocence and goodness alive. This, of course, is easier said than done.

The meaning of the poem is that the things that are most beautiful in life are also the things that have the shortest lives.  He says the first leaves of nature are their most beautiful and that Eden was the most beautiful place to live.  Neither of these beautiful things survived.

I think the clearest illustration of this in the novel is Johnny's death.  He was gold for a brief while, but then died.  On a more hopeful note, it seems that maybe Ponyboy will be different.  Maybe he will actually stay gold.  We get this hope at the end of the book because (instead of going and fighting or anything like that) he turns to writing as an outlet for his emotions.

So Johnny was not able to stay gold, but maybe Ponyboy can.

Frost's poem celebrates the purity of life at its beginning ("nature's first green is gold"), and laments its impermanence ("nothing gold can stay").  When Johnny tells Ponyboy to "stay gold", he is referring to the innocence the greasers once had in common as children.  As they grow into adulthood, they lose that guilelessness and become hardened and jaded under the relentless pressures of poverty, social pressure, family instability, and violence.  Darry, whose dreams of attending college on a scholarship are dashed by the responsibility of caring for his brothers, and Dally, who has given up hope for a better life and descended into a cycle of violence and crime, are bitter examples of this, and the other greasers are not far behind.  Ponyboy is different.  He is sensitive and perceptive, and can still recognize and appreciate the beauty in a good book or a sunrise.  Johnny hopes Ponyboy can keep this ability - that he can "stay gold".

Frost’s short poem “Nothing Gold Can Stay” addresses the fragility of nature, that its cycle is such that green leaves do not last forever: “leaf subsides to leaf” (he puns on the second “leaf,” so that it can be read as “leafs leave”) just as “Eden sank to grief,” meaning just as the pleasures of Eden eventually ended and humanity was thrust into the world where suffering exists. This is the fact of life that Pony learns: good things in life don’t last. In its discussion of the novel, Enotes points out that the poem speaks back to the images of sunsets in the story: “Sunsets are short….But it is possible, Pony proves, to remain true to one's self and thereby ‘stay gold.’”

It refers to Robert Frost's poem "Nothing Gold Can Stay" and has a profound impact on Johnny, who relates it to his own imperiled youth. Later, as he is on his deathbed, Johnny's last words are, "Stay gold, Ponyboy. Stay gold...."

Here is the text of the poem:

Nature's first green is gold, Her hardest hue to hold. Her early leaf's a flower; But only so an hour. Then leaf subsides to leaf. So Eden sank to grief, So dawn goes down to day. Nothing gold can stay.

Basically the meaning of the poem is that innocence is something very hard to hold on to.

The poem from The Outsiders that you refer to is, of course, Robert Frost's "Nothing Gold Can Stay." It is one of Frost's most famous poems, and the book in which it was originally included (New Hampshire) won the 1923 Pulitzer Prize for Poetry. Basically, the poem represents the experiences of life and how the innate goodness and innocence of youth is often hard to hold. The poem includes references to the Garden of Eden, where its innocent inhabitants, Adam and Eve, fell victim to the original sin.

In Johnny's farewell letter to Ponyboy at the end of the final chapter, the dying greaser explains the poem to his friend.

You're gold when you're a kid, like green. When you're a kid, everything's new, dawn. It's just that when you get used to everything that it's day. Like the way you dig sunsets, Pony. That's gold. Keep that way.

The poem is used in part as a reminder to Pony that he is still young and has a long life ahead of him--one which he can still control. Johnny hopes that there is still time for Dally to turn his life around as well, but unbeknownst to Johnny, Dally has already decided that life is no longer worth living.