Aided by the alliteration in b (birds – bees), the phrase (the) birds and (the) bees denotes the facts about sexual reproduction, especially as explained to a child.

When this phrase appeared, birds and bees had long been commonly paired in literary allusions. For example, on Friday 2^nd December 1887, The Indianapolis Journal (Indianapolis, Indiana) mentioned:

James Whitcomb Riley, the Indiana poet, whose homely songs of Western rural life, and rhymes about the birds and bees, have won listeners by their genuine music.

From It’s High Time for a Logical Spring Idea; This May Be the Cure for Fanciful Dreams, published in The Seattle Daily Times (Seattle, Washington) of Sunday 17^th March 1929, the following indicates that the pairing of birds and bees had become a literary cliché:

In the spring a young man’s fancy may turn lightly toward something or other and the flowers may come up, go bloom, but this writer, for one, isn’t going to be tricked into any romantic platitudes about the months of March, April or May, or any other month, for that matter. And if this be reason, make the most of it!
It’s high time that someone in the community took a sane logical view of spring and put a stop to this sentimental twaddle about the birds and bees making hey hey while the sun shines. What with the Mexican revolution, Lindbergh’s engagement, and the Department of Streets and Sewers looking around for another street to dig up, there is quite enough already to think about without running off to the fields to hold hands.

These are, in chronological order, the earliest occurrences that I have found of (the) birds and (the) bees used to denote the facts about sexual reproduction—the earliest texts indicate that this phrase was already in common usage:

1-: From an article by Detlef R. Petersen, United Press Staff Correspondent, published in several U.S. newspapers on Thursday 16^th January 1936—for example in The Wisconsin State Journal (Madison, Wisconsin):

Milwaukee—When your child asks you to explain the mysteries of sex, don’t put him off with vagaries about the birds and the bees.
This was the advice today of Dr. Temple Burling, psychiatric director of the Winnetka, Ill., public schools, who addressed 200 men and women here to attend an institute on sex education of youth conducted by the women’s court and civic conference.
“The child is ready for sex education when he asks for it,” Dr. Burling said. “Parents should answer such questions truthfully and in a natural tone and not put the child off with vagaries about the birds and the bees.
“The social significance of sex needs to be pointed out. The child can’t help knowing there is some force in personality that makes people do strange things.”

2-: From the column Rude Interlude, published in The Chalk Line: Official Organ of the Student Body of the State Teachers College, Johnson City, Tennessee (Johnson City, Tennessee) of Friday 24^th April 1936:

It was Mary Leah who was so fussed because she missed the first lecture demonstration on the facts of life . . . It was this lecture that included the usual story about birds, and bees and boys.

3 & 4-: From The Daily Northwestern: Official Publication of over 12,000 Students (Evanston, Illinois):

3-: Of Tuesday 23^rd March 1937—in Anything Goes: A Woman Replies, by Bobette Kobey:

Gad, Mr. Van Dyne, quit yapping about the women. What’s the matter, Ed, do you have some kind of a phobia? Always howling about how dumb the women are and how unbeautiful they look when they get up in the morning. Well, you can shut up right now, for the biggest hoax ever perpetrated in history was that played by women on men.
[…]
But to explain the Great Hoax to you I must tell you all about the birds. Don’t tell me you know all about the birds and the bees, you just sit tight and listen.
Biologically Speaking
If you have noticed that God made all male birds of larger size and more brilliant plumage than the female. Women birds were always gray, drab creatures which is a shame, but that isn’t the point.
[&c.]

4-: Of Thursday 1^st April 1937—in Campus Diary, by Ed Van Dyne:

Apologies
That Miss Bobette Kobey was offended by my thoughtless remarks concerning women written purely in the spirit of jest in this column on Thursday, March 18 is distressing. The writer of that ill starred article wishes publicly to proffer his apologies.
[…]
You hint, Bobette, at the secret of the birds and bees. Sometime would you tell me more? Ever since I was a little boy, people have said things about the birds and bees and it has me confused. Does it have something to do with the flowers? If so, what kind of flowers?

5-: From the column Smoke Rings, by Earle Dennis, published in the San Francisco Chronicle (San Francisco, California) of Friday 23^rd July 1937—Earle Dennis writes about a film he saw at the cinema:

The opener was a drama of the hills. A simple, lovely maiden of the mountains, with cow-eyes and a sun bonnet, a drunken father, a progeny-emitting mother, and a job in a rural school was about to be done in by a city slicker with a city editor’s conscience. […]
[…]
As the play progressed we discovered that the cow-eyed dame was really quite a cinch. How she had lived to the ripe age of 20 without landing in the detention home was a mystery. She didn’t even know about birds and bees. All the city slicker had to do was to promise to get her sixth brother out of jail, where he was being held on a counterfeit charge, and she threw away everything she owned but her pumps.

6-: From the column A Little O’ This ’N’ Much O’ That, by ‘Bill B. Beaumont’, published in the Beaumont Journal (Beaumont, Texas) of Monday 2^nd August 1937:

I confess a curiosity concerning the whereabouts of Deacon Will Hayes ¹ and his alert censors during the filming of Jean Harlow’s last picture ². Did I state that the episode, wherein the guileless young suitor is informed about the birds and the bees and the pollen and other facts of life, was slightly underdone. Yeah, going pure in my old days.

¹ William H. Hays (1879-1954) was then the President of the Motion Picture Producers and Distributors of America (MPPDA), which had promulgated the Motion Picture Production Code, a set of guidelines for the self-censorship of content, applied to most U.S. motion pictures released by major studios from 1934 to 1968.
² Jean Harlow (born Harlean Carpenter – 1911-1937) was a U.S. film actress and sex symbol.

7-: From the column This and That, by J. P. H., published in The Hutchinson News (Hutchinson, Kansas) of Tuesday 17^th August 1937:

There may be a moral in the story of the 26-year-old Oklahoma boy who shot his pappy because the latter hadn’t told him the facts of life. It is that by the time a boy is old enough to vote he may safely be told about the birds and bees and flowers without thereby getting any wrong ideas.