Nixon:
When they look at you, they see what they want to be.
When they look at
me, they see what they are.
Throughout the film, Nixon compares himself to John Kennedy. He lost the election to him by such a small margin that it was basically half the country for Nixon and half for Kennedy. When Kennedy was assassinated, Nixon seemed upset that not only did a person die, but also his rival died, whom he had great respect for. And at the end of the film, with Kennedy dead and Nixon about to commit political suicide, Nixon mutters those few lines and you could almost get an image of Nixon and Kennedy, two complete opposites, enemies, Presidents with their terms cut short, shaking hands in admiralty of each other.
