To build from what pohnpei397 has explained, we might begin by generalizing and say that this is a love poem. The poet is writing about the pages of poetry that he is writing (leaves = pages) and which he intends to give to his beloved.
The poem's rather dramatic and even fatalistic tone is expressed early and often as the poet contrasts the happiness and blessedness of the pages of poetry when they are in his lover's hands with the "dead-doing" and "sorrows" of his "dying sprite." Even as his spirit dies, he hopes that there will be some (imagined) happiness for him when his lover reads the poems he has written for her.
Overall the poem seeks to express a devotion to a lover and to articulate the great esteem the poet holds for her. She is compared to an angel and the implication is made that she possesses the power to bestow happiness.
Given the meanings discussed here, the poem naturally fits into Spenser's scheme for this book of sonnets, "Amoretti (meaning 'little love gifts' in Italian)" (eNotes). A poem of praise and devotion, Sonnet #1 begins the cycle of little love gifts that make up the larger work.
The first quatrain is saying that he envies leaves (I think he's using it in the sense of pages) that his love holds lovingly in her hands -- hands which hold his life (because he loves her so much.
The second quatrain is saying that the lines he writes should be happy because his love will look at them. Even if she's crying for his death, at least the lines should be happy because she's looking at them.
The third quatrain is saying his rhymes should also be happy because she's looking at them. And he's saying that she is his muse -- Mt. Helicon is where the muses were supposed to be from.
The couplet says that all those things should only care about pleasing her because he himself doesn't care about anything else.