The reason is that the field of stars is to be worn closest to your heart. Moreover, the flag should look like it’s advancing, not retreating. So, if a patch is worn on your right sleeve, you should apply it backwards.
Tim Marshall, author of "Worth Dying for: The Power and Politics of Flags states:
"Just as the US flag dips to no man or king and you will see even at the Olympic ceremonies, the American flag is the only one that doesn't dip to the head of state of the host country. Because — it's not a mark of disrespect to them, it's a mark of respect to the American flag. And they take it so seriously that it must always face forward.
Now, on a flag pole that puts the stars on the left-hand side next to the flag pole, that's the most prestigious position. On an arm patch, you are looking at it differently and when the soldier, or marine or whatever, marches forward, the US flag most face forward. It must not be seen to be in retreat. And so the stars are actually now on the right-hand side of their badge and so they face forward, just as it never retreats. It's always in its special position when it's flown on a car.
You might think this is taking things to extremes but when you really get to the bottom of flags, they are about extremes of passion and extremes of belief. And the Americans take their flag very, very seriously."
Other interesting flag facts:
