The meanings were added, not exactly entirely substituted. The pagan version constituted something on the lines of "rebirth", "regeneration" of Nature's cycle. A sense of renewal of hope, which aligned itself with the solstice, the moment the sun began to have a bigger presence than the day before (at least in the northern hemisphere). Christendom interpreted this moment as the moment when Jesus was born, which then symbolizes every child born within the families, which again is a symbol of renewal of generations and of hope of a better future, etc., etc.
I do prefer christian's version. It's more anthropogenic.