Ok first of all, its a a rhetorical question. Which means hes asking himself if another person a question to which he knows the answer. That being that yes he does of course "destroy" them as enemies.
I dont believe Lincoln actually believed he was "destroying" his enemies by befriending them. From a philosophical point of view he is right. Think about it. Lets say you and i have a punch up we used to be friends we are now enemies cos i stole some money from you and we had a punch up etc.
for years you tell people how much you hate me, you want to see me fail, when you see me in the street you chase me and i see you as my enemy correct?
Well lets say i bring you a gift one day, i bring you your money back along with some flowers, and a sorry note. Just this small act should turn you around even in the slightest so much so that you are no longer my enemy because i have befriended or tried to befirend you. Do you see how this takes away your powers as an enemy?
Its like the old saying "If you cant beat em join em".
In spirituality they teach that whatever you focus on will grow. We know that in this world everything is made of energy and energy at its core is a vibrational frequency. Thoughts are energy so they are also frequency. Well by focusing on a person as an enemy you are attracting that energy because why? Well watch the secret... but in short frequency attracts other frequencies of the same kind. Example, you never so bad people hanging with priests and stay at home mums do ya?
All in all what Lincoln was trying to say is that he is destroying the nature of the enemy by befriending them. How can your friend be your enemy? Its not possible.