Yeats chose to locate "The Circus Animals' Desertion" in a circus for a reason: it draws on an old tradition of using the carnivalesque to describe dreams and fancies that are alternatively bizarre, ephemeral, and thrilling.
Mikhail Bakhtin, a Russian literary critic in the early 20th century, used this term to describe anything that undoes traditional structures of power and creates space for the subservient or the non-dominant classes and identities to come into the light. The carnivalesque is defined by transgression, usually bodily, and is a location where social norms can unwind.
In medieval times, the carnival was a place where the confines of the church and state could be transgressed. They allowed for free thought and revelry and, with their freak shows and acrobats and music and noise, could catalyze the sort of free-thinking abandon that created movements such as the Renaissance. The carnivalesque can be seen in modern counter-culture movements, such as the American beatnik movement of the 1960s and hippie movement of the 70s, which also created worlds in which deviant identities could be seen, in which capital was not the center of culture, in which love and art were central forces, not war and hatred.
But these movements, and the carnivalesque itself, are always ephemeral, and always seem to be swallowed up or dominated by larger forces. This is the nature of the counter-culture and the carnival: they are, by definition, opposed to the mainstream.
Yeats' lifelong unrequited love Maud Gonne probably would have been some sort of revolutionary or member of the counter-culture in this century; she certainly gave herself up to her cause, fighting for Irish freedom against the looming, constricting force of dominant England. Yeats' love for the carnivalesque and for revolution perhaps belie the fact that he himself was never able to fully subscribe or participate in these revolutions. Yeats was also drawn to the occult, another counter-culture movement that broke the rules of the church and believed that man could unravel the mysteries of the universe.
Where Maud Gonne believed firmly in nationalism, Yeats' poetry seems to presage the postmodern movement, which, 40 years after Yeats' time, would theorize an absence of center and meaning. Essentially, postmodernism would view all life like the raggedy street of "The Circus Animals' Desertion"'s final stanza. It also proposes that community and purpose are illusions, simulations of a sort; however, it is not always entirely nihilistic. There is room to create, some postmodern theorists write. And so the circus is reborn, at first ramshackle and threadbare, sewn out of nothing but the rags and bones at the bottom of the heart, growing until it is time to go again.