When you have no companions, this perk grants a 10% increase in both the damage done by the Courier and your chance to hit in V.A.T.S. To use this perk, companions must be dismissed; telling them to wait will not grant the perk's bonuses.
Behind the scenes
The representation of the Lonesome Road perk is a reference to the Vault Dweller ending in Fallout, the Lone Wanderer ending in Fallout 3 and the Courier ending in Fallout: New Vegas, where the three are shown walking away, with their back turned and showing the number of the Vault jumpsuit they wear. In the Fallout 3 ending, the Lone Wanderer is shown with Dogmeat while in Fallout and Fallout: New Vegas, the character is shown alone.
This perk is not always given to the player, and is un-fixable unless Lonesome Road is re-installed.