The Bee Box is simply a house for bees. By catching several Bees with a Bug Net and getting the Honeycomb from a smashed Bee Hive, you'll pretty much be ready to build your own Bee Box. Every morning, the bees will come out and visit the nearby flowers. As they visit the flowers, they'll deposit honey in the Bee Boxes. At night, they'll head back to the bee box to sleep.
Getting the Honey from the bee box can be a little tricky but if you have at least a Log Suit and Beekeeper Hat then you should be pretty safe. Wait for the four bees to leave the Bee Box, run up to it and harvest. Run away before they return. If they are far enough away, you shouldn't have a problem. Trying to harvest the Honey at night can be a little bit more dangerous since the bees are already in the box to begin with. They'll likely manage to sting you at least once before you get away.