Because saying “Happy Holidays” to a stranger is not “taking Christ out of Christmas”; instead, it’s acknowledging the plurality of ways that late December is treated in a world in which not everyone is Christian. “Happy Holidays” is a catch-all that includes Christmas.
Christians, at this time of year, celebrate the (supposed) birth of Jesus; Jews, the (supposed) lamp oil miracle of Hanukkah; secularists, relaxed family get-togethers; sports fans, football; mall rats, sales. Each of these are holidays of one sort or the other to different people—breaks from the cycle of work—and thus much of what goes on at “Christmas” has nothing to do with actually remembering Jesus’s birth.
So happy holidays, one and all.
Prometheus Unbound 
My contribution:
You know, here in Europe everyone says “Merry Christmas” and no one of other religions object, not even Muslims if you can believe that. The American approach seems like avoiding latent conflicts by walking on eggshells.