I have read many articles on this topic and below is the most common explanations, but as with every wedding superstition, carrying the bride over the threshold has many variations of the following.
The threshold superstition actually dates back to ancient Roman cultures. People believed that brides were susceptible to evil spirits through the soles of their feet, and many of those evil spirits liked to hang out on the threshold of homes. The groom carried the bride across the threshold to make sure that she didn’t get attacked by any of these spirits.
Other variations on this superstition’s origin exist, too. In Medieval Europe, grooms carried brides across the threshold to avoid onlookers seeing the bride as scandalously eager to consummate the marriage. In the later days of Europe, some believed that if the bride tripped on her way into the new home, it would cast bad luck on the marriage and home. Hence, the groom would carry the bride across the threshold to avoid any chance of her tripping on her way inside.