I used to work for the postal service, which does something different, but similar. (Also, UPS, FedEx, and the USPS all have strategic partnerships, so we had UPS and FedEx folks on site every night.)

For UPS and FedEx, it's actually cheaper, faster, and easier to consolidate sorting operations under fewer roofs and fly everything to and from these hubs than it is to decentralize those operations and have them everywhere.

UPS and FedEx both have hubs in the US where every package goes to. The buildings are huge -- think of buildings where you could host 2-3 football games under one roof. IIRC, both companies operate two buildings each, and they're both in the Knetucky/Indiana/Tennessee area (usually within about an 8-10 hour drive of at least 50% of the US population.)

So your package gets scanned by the driver, brought to the local outpost, usually located at or near an airport, and stuck on a plane. Overnight, that plane flies to the nearest hub, where the packages are all sorted into destination and service level (next day, 2 day, ground, etc.)

The sorting is done by machine. When you consider that every night, literally millions of packages are being sorted, you realize that it's basically impossible to cost-effectively do all that sorting by hand. There just aren't enough hours in the day, certainly if you want your package tomorrow. The machines are gigantic. The one I worked alongside was probably about half a mile in overall length, maybe more -- it could sort mail into about 1,000 ZIP codes, and had a capacity of about 250,000 packages per day.

Modern sorting machines are capable of nailing down the sort to not just the local outpost, but also to the specific driver. (For the postal service, there are machines that can sort letters into trays such that they're arranged in the order the letter carrier reaches each address on his or her route.)