The speaker begins by using a simile to compare his or her "luve" to a rose (the unfamiliar spelling here is just part of the poem's Scottish dialect). "Luve" here could actually have two meanings. It could refer either to the beloved (that is, the person the speaker loves), or to the speaker's feelings of love for this person. Comparing the beloved to a rose emphasizes her youth and beauty, while comparing the speaker's emotions to a rose emphasizes how intense, exciting, and new those feelings are.
The rose is also a key symbol in the poem, in two different ways. On the one hand, the rose is an ancient symbol of love in many cultures, including in Western literature. Different colors of roses have different symbolic significance; the color red is associated with true love. By using this common image to describe his or her love, the speaker frames the experience of love in universal terms, inviting the readers to recall their own experiences of love as a way to understand the speaker's.
The rose has another significance besides love, however. Flowers often symbolize impermanence, since they are so short lived. A "newly sprung" rose is especially short lived since its newness, by definition, lasts only a short time. Instead of symbolizing the intensity of the speaker's love, then, the rose may possibly signify that these feelings of love may only last a little while. The reader must continue through the rest of the poem to see which interpretation of the rose is the correct one.
The first two lines also establish the poem's meter, which is alternating iambic tetrameter and trimeter (meaning lines alternate between have four and three iambs—poetic feet with a da DUM syllable rhythm—per line):
O my Luve is like a red, red rose
That’s newly sprung in June
The poem follows this meter throughout, though with some irregularities. In the first line, for instance, there is an additional syllable the start of the line for the interjection "O." There is also a spondee with the words "red, red," which can be read as both being stressed. The spondee, like the repetition of "red" and the alliteration of "red, red, rose," emphasizes the brightness and vividness of the color of the rose by emphasizing the sound of the phrase.