The name derives from the tradition of giving seasonal gifts, on the day after Christmas, to less wealthy people and social inferiors, which was later extended to various workpeople such as labourers and servants.
The traditional recorded celebration of Boxing Day has long included giving money and other gifts to charitable institutions, the needy and people in service positions. The European tradition has been dated to the Middle Ages, but the exact origin is unknown and there are some claims that it goes back to the late Roman/early Christian era.
In the United Kingdom it certainly became a custom of the nineteenth century Victorians for tradesmen to collect their "Christmas boxes" or gifts in return for good and reliable service throughout the year on the day after Christmas.
The establishment of Boxing Day as a defined public Holiday under the legislation that created the UK's Bank Holidays started the separation of 'Boxing Day' from the 'Feast of St Stephen' and today it is almost entirely a secular holiday with a tradition of shopping and post Christmas sales starting.
Here in the US, at least in my part of it, Boxing Day has become the Black Friday after Christmas. The stores are just as packed today as they were on Black Friday. Not me, I hate shopping, and shopping on a day when everyone else is shopping is at the bottom of my list. I’m usually at a store when it opens, avoiding as many other shoppers as I can. Nope, no shopping for me! There is nothing in any store that I need that bad.