The separation here is not a literal trip. Instead, Burns is using the traditional idea of separation from a love to underscore the tenderness and purity of the speaker's feelings for his beloved.
He describes it as the freshest of roses "newly sprung in June", and a "melodie sweetly played in tune". The latter, of course, is a nod to the Scottish oral tradition on which Burns drew in writing his love poetry. The speaker then describes the depth of his love as as fair as his beloved herself. Here again, the expression of love is very traditional, and goes back well into the oral tradition.
However, Burns's description of the time period over which his love will endure marks him as an Enlightenment poet. He references the geological time recently identified when he discusses the seas drying up, and the rocks melting with the sun. The time he references here is well beyond the human lifespan, and demonstrates clearly the eternity of the speaker's love for his beloved.
It was quite common in Scottish poetry of the eighteenth century to reference a separation. This might be due to the speaker's allegiance to the lost cause of the Jacobites and an exile or it might be due to his forced service in the British army (an aftereffect of the failed Jacobite movement). The use of hyperbole in the final lines, however, underscores the depth of the speaker's love.