There are two popular explanations for this custom. Since the Kiddush, the blessing over the wine, precedes the Motzi, the blessing over the challah, we cover the challah so that it should not feel slighted by the attention paid to the wine. Jewish literature often likens Shabbat to a bride. Just as the veil of a bride is lifted under the chuppah (wedding canopy), the challah is uncovered prior to the recitation of the blessing.
An important custom in Judaism is that of hidur mitzvah, making the commandment pleasing. It is not simply enough to put an unadorned cloth over the challah, but to have or make a beautiful cover that is just for this purpose. Making a challah cover is a wonderful family activity.