O my Luve's like a red, red rose
That's newly sprung in June: (1-2)
The speaker compares his love to a "newly sprung" flower; his love is fresh, new, just beginning its life. But wait. That's kind of weird, right? It seems like his love is very deep, like he's loved his bonnie lass for quite some time. So how can it seem so new?
O my Luve's like the melodie
That's sweetly play'd in tune! (3-4)
His love is not of the stormy variety. Nope, it's all hearts and flowers and melodies. He's all swoon, all the time.
As fair art thou, my bonnie lass,
So deep in luve am I: (5-6)
The speaker says he is as "deep in luve" as his "bonnie lass" is fair. What if, however, the "lass" isn't really that "fair" (i.e. pretty)? If she isn't that good looking, then he really isn't that in love. This isn't a very likely prospect, but the language certainly leaves the possibility open. We just couldn't resist pointing it out.