Simply kill any Legion soldiers and bring back the Legion ears to turn in.
The objective is to kill as many Legion soldiers as possible. The best way to do this is to activate the quest, and then do Restoring Hope. The Courier needs to turn in at least 30 ears for Private Sexton to give them a reward of 100 caps and 6 beers. After the reward has been given, one can no longer collect Legion ears.
There is bonus applied to the third part of Restoring Hope. Every 10 ears turned in adds a morale bonus of 1 for a maximum of three. This bonus is applied to the Nelson attacking forces.
After one has turned in enough ears, they will also receive a number of beers and 100 caps.
Due to Sexton not needing to be killed in We Are Legion, it is possible to give him ears for reputation, even if the rest of the camp is dead.
The number of ears collected will be represented as tally marks in the mess hall. There is also a bull with the initials of Caesar's Legion above it. These marks stop at 30 ears, no matter how many have been turned in.
With few exceptions, such as the dead Recruit Legionary in the booby-trapped Nipton Home, ears won't start appearing on Legion corpses until speaking with Private Sexton.
Various Legionaries— including those at the Fort, the Legate's camp, members of Legionary assassin squads, and those that attack Bitter Springs during Boone's quest— do not drop ears upon death. For this reason alone, it is probably best to speak to Pvt. Sexton prior to completing Restoring Hope, in order to maximize one's "ear harvesting" opportunities.