"Not because they were deranged, not because they were nuts," Biden continued, according to a transcript. "Because they'd been to the top of the mountain, and they just knew in their heart they'd never get there again, that it was never going to get - never going to be that way ever again. That's how an awful lot of you feel."
In 1972, just after the Delaware Democrat was first elected to the Senate, his wife, Neilia, and his 13-month-old daughter, Naomi, were killed in a car crash. Biden's two sons - Beau, then 3, and Hunter, 2 - were grievously injured but survived.
On Friday, Biden told the military families how low the crash had brought him. "I probably shouldn't say this with the press here, but - no, it's more important - you're more important," he said.
Biden had actually told the story before, on page 80 of his 2007 memoir, Promises to Keep.
"I began to understand how despair led people to just cash it in," Biden wrote.
On Friday, that story was a powerful section of a speech that illustrated Biden's particular style of rhetoric: frequently meandering, slightly pompous but movingly personal.
Biden often veered from the topic at hand - once, to tell the story of how he proposed to his current wife, Jill, five times before she said yes.
And, in a manner as unpolished as a living-room conversation, Biden told of climbing back out of grief.
"I have to tell you, I used to resent - I knew people meant well. They'd come up to me and say, 'Joe, I know how you feel,' " Biden said. The audience laughed.
"Right?" They clapped.
"You knew they meant well. You knew they were genuine. But you knew they didn't have any damn idea how you felt," Biden said. Laughter. "Right? Isn't that true?"
Biden talked about his internal conflicts, as he tried to start another relationship after his wife's death.
"You're going to go through periods when, after a while, you'll see somebody you may have an interest in, and you're going to feel guilty as hell. You're going to feel this awful, awful, awful feeling of guilt," he said.
Biden did not look like a vice president giving a speech: He hunched over, he looked down at his hands, he spoke at times haltingly and at times through clenched teeth.
And he told the story of his slow recovery - relying on family members and calling other people who'd been through the same kind of loss.