I would say that the theme of this song or poem is the idea that love is what makes the world go 'round, as they say. The speaker is saying that it is "the lasses" that make life happy.
In the poem, the speaker says, for example, that the material things in life are no good when compared the pleasures of being in love. He says that you can have all the riches you want, but you will not really be content. By contrast, if you have a love, if you have a woman to spend your time with, you will be a happy person.
Eighteenth century Scottish poet and lyricist Robert Burns was well-known for his affinity for women. And that affinity was quite explicit in some of his poems and lyrics. Take, for example, his poem unsubtly titled The Fornicator:
For I've lately been on quarantine,
A proven Fornicator.
This affinity for women constitutes the theme of his lyrics to the song Green Grow the Rashes, a potential euphemism if ever one existed given the theme of The Fornicator. Whereas the latter is explicit in its emphasis on sex, however, the former is more a paean to women. Green Grows the Rashes is a celebration of women. A man well-accustomed to wine, women and song makes very clear early in his song that the dominant occupation of his life is women:
The sweetest hours that e'er I spend,
Are spent amang the lasses, O.
Burns' lyrics are replete with references to women and his preference for being in their company. Each stanza is a reaffirmation of that sentiment. Furthermore, he clearly emphasizes his preference for the company of women over that of his fellow males, as when he writes,
For you sae douce wha sneer at this,
Ye're not but senseless asses, O!
The wisest man the world e'er saw,
He dearly loved the lasses, O!
The theme of Green Grow the Rashes is the author's love of women. He treasures his relationships and he certainly treasures his sexual experiences, drunk or sober. Burns remains renown for his proclivities in that regard, and his reputation in his native Scotland remains unblemished.
The speaker is an individual who is celebrating his love for women. He is unabashed in his pronouncements and there is no reason to take his remarks as anything but serious, despite his hyperbolic praise. The speaker claims that he has spent his “sweetest hours” among “the lasses.” He also seeks to corroborate his attitude by claiming that the “wisest man the war’l e’er saw” was also a person who “dearly loved the lasses, O.” He is uncomplimentary toward sober people who might sneer at his pronouncements (lines 17–18). This view of men professing love is a natural extension of the poet's feelings of expression with no holding back.