The speaker says that his love is "like a red, red rose, / That's newly sprung in June." This is what's called a simile, a figure of speech that compares one thing to another of a very different kind using the word "like" or "as." It's quite common in love poetry to use the rose as a symbol of love. Burns also emphasizes the passionate nature of that love by repeating the adjective "red." This isn't just any old love; this is love of a particularly burning intensity. And the passion with which he's been seized also seems to have come out of nowhere; it's sprung up suddenly like a new rose in June.
The speaker then goes on to describe his love as being like "the melodie, / That's sweetly play'd in tune." Once more, the intensity of this love is emphasized. There's nothing quite like a sweet melody to stir the passions. Crucially, however, this melody's played in tune, implying not just the sweetness of the melody but also the harmony that exists between the speaker and his lover.