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One of the points that gets repeated about the White Witch's power is that she has made it always winter in Narnia, but never Christmas. (Presumably because she hates fun, but maybe just because she hates ugly sweater parties? We'll never know for sure.)
The appearance of Father Christmas (or, as Americans would call him, Santa Claus) is a sign that the Witch's power is weakening. It's important to notice that C.S. Lewis's conception of Father Christmas is more awe-inspiring and than our idea of jolly old St. Nick. Unlike the fat man in the red suit who likes to eat cookies and give toys, Lewis's Father Christmas is "so big, and so glad, and so real" that he makes everyone feel "very glad, but also solemn" (10.35).
If you think of the Father Christmas in this book as a cross between Santa Claus and a Celtic god of the wood, then you might have an idea of what Lewis was going for—after all, as a scholar of medieval literature, Lewis knew a thing or two about early British mythology.
The power and solemnity of this version of Father Christmas are reinforced by the type of gifts that he gives. Instead of distributing toys and games and weird sweaters, Father Christmas gives Peter, Susan, and Lucy tools—in fact, weapons—that they can use in the upcoming battle between Good and Evil.
But he also spreads cheer, giving the Beavers and children a delicious tea, and giving a feast to another party of animals and creatures in the woods.