As in, the analogy describing reproduction.
It seems that the phrase has multiple possible sources with none truly confirmed. I've listed the sources below:
"..poet Samuel Coleridge (1825): 'All nature seems at work ... The bees are stirring—birds are on the wing ... and I the while, the sole unbusy thing, not honey make, nor pair, nor build, nor sing.'"
"earlier instances of this idiomatic expression appear in the Cavalier poet, Thomas Carew's work, "The Spring" (c.1640), in which, Carew uses earth and its change of seasons as a metaphorical depiction of women and their sensuality"
"Dr. Emma Frances Angell Drake (b. 1849) wrote a section of a publication called The Story of Life which was published in 1909 ... "when you discovered the tiny blue eggs in the robin's nest and I told you that wrapped in each shell was a baby robin that was growing there, kept warm by the mamma bird..." the narrative continues on in vague terms without actually describing sexual intercourse."
"Several sources give credit to Cole Porter for coining the phrase. One of the musician's more famous songs was 'Let's Do It, Let's Fall in Love.'"
These all come from this Wikipedia page. But it seems that there will never be a hard and true answer to this question.
There seems to be no certainty about where the phrase originated from but the song certainly could not have originated the phrase since it plays off of the phrase. It uses an allusion to the "birds and bees" phrase to make the listener think of something risque then revealing that it's actually referring to "falling in love." That's the charm of the song. Like a reverse double entendre. This would not work if the listener was not already familiar with the popular phrase.