In this passage, Christ addresses the Roman practice of commandeering civilians or their property (mules, horses, oxen, camels, carts, wagons, etc.) to carry the luggage or other burden of military personnel for, in this case, one mile.
Evidently, the practice did not originate with the Romans but with the Persians. As there were no post offices at the time, and in order that royal orders might reach their destination quickly, Cyrus set up a system not unlike our Pony Express. A rider in this service was empowered to take a civilian's horse (usually his best or only horse), if his was worn out or lame. In addition, he could press a boat, cart, or any other vehicle into the king's service.
In recent centuries, this practice, often used to force seamen into the service of another nation's ships, has been called impressment. In America's Revolutionary War period, British ships would often intercept other nations' ships and force any American sailors found on them to work for the Royal Navy. In Roman times, a man could have worked all day, his family waiting for him to come in from his fields, and suddenly, a Roman soldier could order him to carry a heavy load for a mile.
No one likes to be made to do someone else's work. At the very least, we are apt to complain, argue, or simply refuse to be so used. Being compelled to engage in "community service" by law or by might is demeaning and perhaps unjust. But Jesus tells us to take the sting out of the situation by being willing to carry such a burden an extra mile in a cheerful attitude.
In a similar vein, Solomon advises, "If your enemy is hungry, give him bread to eat; and if he is thirsty, give him water to drink; for so you will heap coals of fire on his head, and the LORD will reward you" (Proverbs 25:21-22). Jesus says something very similar in His subsequent teaching (Matthew 5:44-45). Being struck, sued, or forced to carry a heavy load can bring out the worst in human nature: anger, resentment, outrage, and even violence. But when those who have been called find themselves in difficult and trying circumstances, their attitude must not be belligerent, spiteful, or vengeful, but helpful, willing, and good-natured. "Above and beyond" must be their motto.