Ancient Romans held a superstition that a newlywed couple could be plagued by misfortune, if the woman tripped in the entrance to their home. To prevent this from occurring, the groom would carry the bride inside.
(Edited to Add: I so would have tripped, so thank you Ryan!)
Putting the Right Foot Forward
Early Slavic societies nurtured the belief that a bride must not set her left foot into the newlyweds' home before her right one. Apparently, this would result in many years of bad luck. By entering the home in her beloved's arms, she would not be exposed to this risk.
Shielding from the Nether World
Greek mythology carried stories of evil spirits living in the earth, particularly underground. For this reason, young maidens would scatter flower petals in a bride's path for the wedding celebration. Afterwards, when the couple would retire to their home, the groom would carry the wife through the door to protect her from the evil netherworld spirits underground.
Blocking Demons' Access to the Home
The Viking peoples suspected a bride might unwittingly carry demons from her household of origin into her new home. If her bridegroom lifted her initially into their home, the demons would be prevented from entering with her.
(Edited to Add: I don't think I brought demons with me, but nevertheless, they're gone if I did!)
Dragging the Maiden into the Cave
Of course, we have all heard stories of the cave man, who would growl and grab a woman by her long hair before dragging her into his cave. Ugga-bugga! Call it progress! What woman wouldn't rather be swept off her feet than snatched by her hair by some big lug?