It's sort of an odd little song to have become a hit, and especially to have been recorded by Peggy Lee. The tune is very simple, by Cy Coleman, and Carolyn Leigh's lyrics are brief and tend to get repeated. But it's very infectious, and a song I always loved as a kid. (For that matter, Frank Sinatra recorded the song, as well.)
I thought it would be fun to post two versions here. Peggy Lee's recording, as I said, is the one that's best known, but here it is from the movie, in the opening title sequence. The movie takes place during WWII, and it's being played over the tinny radio, so the style is set to fit that era. The singer is Digby Wolfe.
By the way, you'll note in the credits that the movie is directed by Ralph Nelson. Father Goose was reasonably successful, but he directed an even more gem of a movie the year before -- Lilies of the Field, for which famously Sidney Poitier became the first black man to win Best Actor. And one other note: the film of Father Goose was written by Peter Stone and Frank Tarloff -- both had long careers, but Stone is particular has been well-represented on these pages as the writer of the Tony Award-winning Broadway musical (and its film adaptation), 1776,
But this is about Carolyn Leigh, so let's not let her pass us by...
And then, just so's you can hear the song in its richer, and far-better arranged version, here's the pop hit recording by Peggy Lee.