What is the significance of the falconer?

The falconer in "The Second Coming" is generally thought to represent Christ. The Christian historical epoch, or "gyre" as Yeats calls it, is drawing to a close. In its stead will come a new era marred by chaos, bloodshed and disorder. The figure of the falcon in the poem represents man and the civilization he has built. But because of the gyres' constant turning, the gap between the old and the new is widening, so much so that we're becoming separated from Christ. Because we are moving away from Christ, we no longer heed his message. This is what Yeats means by "The falcon cannot hear the falconer."
The falconer also hints at Yeats' fundamentally aristocratic understanding of politics. Hunting with falcons is an activity traditionally associated with the upper-classes, with "the best people" in society. For thousands of years they have been in charge of things, but now their domination is under imminent threat, about to be swept away by the coming of the new era. "Mere anarchy" is about to be "loosed upon the world," heralding the dawn of a new democratic age in which the "best (i.e. the aristocracy) lack all conviction" while "the worst" (the unruly mob) are "full of passionate intensity."
Just as we no longer hear the word of Christ, neither do we pay any attention to what Yeats sees as the innate wisdom and good sense of the European aristocracy, characteristics which make them uniquely born to rule.
https://www.poetryfoundation.org/poems/43290/the-second-coming