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.
And I will luve thee still, my dear,
Till a' the seas gang dry: (7-8)
The word "still" is interesting; it can mean both "remaining the same," as in "even after you get old, I will still love you," as well as "motionless" or "quiet" (like a corpse). Not to get all morbid or anything, but it almost sounds like the speaker is saying, "I will luve thee still" in the sense of "I will luve you until you are still and motionless." (Just think of the phrase "I will love you silly.") This interpretation makes the poem seem just a teensy bit darker. Okay, maybe a lot darker.
I will luve thee still, my dear,
While the sands o' life shall run. (11-12)
Lines 11 and 12 have an odd number of syllables (7), which makes them stand out or seem kind of weird in a poem with such an obvious form. But hey, maybe this weirdness could mean that the speaker's passion is getting out of control to the point that he can't keep his verses nice and neat. Or it could reflect some hesitation, too. Maybe he's sweaty-palmed about professing his love.
And fare thee weel, my only Luve,
And fare thee weel a while!
And I will come again, my Luve, (13-15)
In the final stanza, the speaker rhymes "Luve" with… "Luve." Sure, this is kind of cheap (who rhymes the same word in the same stanza, right?). But it does suggest that the speaker can only think about one thing: love. The fact that he says love twice here, and a few other times in the poem, definitely hints that he's got a one-track mind.