The poem refers to the fact that sometimes beautiful things do not last long. They tend to be evanescent. Gold is of course associated with the precious metal. It is extremely valuable in this sense. But in the poem, the meaning has more to do with the color of foliage just prior to when it turns green. This is especially true during spring time in New England (where Frost lived). The speaker says that it is hard for Nature to "hold" the color gold. This just means that the gold hue doesn't last long. The flower or plant "subsides," folds, or fades into green leaves.
When the speaker says that "Eden sank to grief," he is referring to the story in the Bible. Initially, Adam and Eve live in a paradise, the Garden of Eden. However, this doesn't last long because they eat the forbidden fruit. Just as the golden hue of the flower doesn't last long, their access to paradise doesn't last long.
The title "Nothing Gold Can Stay" illustrates how these beautiful things (the gold hue of the plant and the Garden of Eden) do not stay long. However, the resulting green hue does represent life and rebirth. Losing the Garden of Eden is referred to as a "felix culpa" which means a happy fall. It is tragic to lose paradise, but in doing so, Adam and Eve (humanity) gain knowledge of good and evil and free will. So, the fact that a beautiful thing is fleeting is not so tragic because it can (in these two examples) result in new life. This idea also ties into the seasons and the cycle of life.