The phrase often implies disapproval or rejection of the person being sent away. After cheating in the game, the player was sent packing.
Pro Tip 2/3
Informal Tone
Used in casual conversation, it's not suitable for formal contexts like business or diplomatic discussions. He was always late, so the boss sent him packing.
Pro Tip 3/3
Authority and Power
The person doing the 'sending' has authority or power over the other. The teacher sent the unruly student packing from class.