I believe that Burns is trying to put in our minds the image of a lover who is declaring his love for his beloved. However, I also think that Burns is intending to create some question in our mind as to how serious the speaker is.
The first three stanzas of the poem show us the first of these images. The speaker uses a few similes and some hyperbole to tell his beloved how much he loves her.
But then in the last stanza we see that he is leaving her. Is he just saying this stuff to make her feel better? Or does he really mean it? It's impossible to know.
Towards the end of his life Robert Burns (1759-96) was engaged in the task of collecting old traditional scottish songs. In one of his letters in 1794 he states that the song "A Red, Red Rose" is "a simple old Scots song which I picked up in the country." Hence the poem is not his original composition.
Burns' poem, however, is charmingly simple and direct in its method of praising his lover and most significantly describes how much he loves her:"As fair art, thou my bonny lass/So deep in luve I am."
He then tells her how much he loves her:
I will luve thee still, my Dear,
While the sands o'life shall run.
The implication is that he will love her forever, that is, infinity. As long as human life exists on this earth he will love her. Burns uses hyperbole, that is, exaggeration to convey to his lover the depth and intensity of his love for her. In the previous line he has told her that he will love her till all the seas dry up! But he is not satisfied with that, because he feels that there is a possibility that all the seas may indeed dry up so he says that he will love her till all human life comes to an end on planet earth!
Robert Burns is creating an image of love that endures until the end of time. He uses his lyrical prose in the same manner as a song of love to his beloved. He identifies that death comes but that love can continue on without the presence of his beloved.
In the poem he compares his love to a red red rose. Roses are symbolic of several things: beauty, endurance, and strife. Roses have great beauty but also flaws. Their thorn can stick you. Burns uses the idea of roses as a symbol of the person he loves. Her flaw is that she will grow old, beauty will go away much as it does in a rose, and she will die, which can be the final thorn. However despite these things he will continue to love the girl forever and even death can not end his love for her.