A few theories.
Centuries ago, ordinary members of the merchant class gave boxes of food and fruit to tradespeople and servants the day after Christmas in an ancient form of Yuletide tip. These gifts were an expression of gratitude to those who worked for them, in much the same way that one now tips the paperboy an extra $20 at Christmastime or slips the building's superintendent a bottle of fine whisky. Those long-ago gifts were done up in boxes, hence the day coming to be known as "Boxing Day."
Christmas celebrations in the old days entailed bringing everyone together from all over a large estate, thus creating one of the rare instances when everyone could be found in one place at one time. This gathering of his extended family, so to speak, presented the lord of the manor with a ready-made opportunity to easily hand out that year's stipend of necessities. Thus, the day after Christmas, after all the partying was over and it was almost time to go back to far-flung homesteads, serfs were presented with their annual allotment of practical goods. Who got what was determined by the status of the worker and his relative family size, with spun cloth, leather goods, durable food supplies, tools, and whatnot being handed out. Under this explanation, there was nothing voluntary about this transaction; the lord of the manor was obligated to supply these goods. The items were chucked into boxes, one box for each family, to make carrying away the results of this annual restocking easier; thus, the day came to be known as "Boxing Day."
Many years ago, on the day after Christmas, servants in Britain carried boxes to their masters when they arrived for the day's work. It was a tradition that on this day all employers would put coins in the boxes, as a special end-of-the-year gift. In a closely-related version of this explanation, apprentices and servants would on that day get to smash open small earthenware boxes left for them by their masters. These boxes would house small sums of money specifically left for them.
This dual-versioned theory melds the two previous ones together into a new form; namely, the employer who was obligated to hand out something on Boxing Day, but this time to recipients who were not working the land for him and thus were not dependent on him for all they wore and ate. The "box" thus becomes something beyond ordinary compensation (in a way goods to landed serfs was not), yet it's also not a gift in that there's nothing voluntary about it. Under this theory, the boxes are an early form of Christmas bonus, something employees see as their entitlement.
Boxes in churches for seasonal donations to the needy were opened on Christmas Day, and the contents distributed by the clergy the following day. The contents of this alms box originated with the ordinary folks in the parish who were thus under no direct obligation to provide anything at all and were certainly not tied to the recipients by a employer/employee relationship. In this case, the "box" in "Boxing Day" comes from that one gigantic lockbox the donations were left in.