Found Bill Withers' intro to the song here. His full statement is:
Women can say stuff like, "I loved him, I really really loved him. But he just left. Why'd he leave like that?"
Men, given the same situation, usually say somethin' like, "I'm glad the old jive broad split, man," knowing all the time that it's really killin' him inside.
Here's the breakdown of the slang words used:
old = (informal - adj) "Used to express affection, familiarity, or contempt."
jive = (North American - informal - adj) "Deceitful or worthless."
broad = (North American - informal - noun) "A woman."
split = (informal - no object) "Leave a place, especially suddenly."
So he is giving an example of how a man usually hides his feelings after a woman leaves him. That line means:
"I'm glad that old jive broad split" = (disrespectfully) I am glad that worthless woman left me
He finishes explaining that he didn't want to be like that, so he wrote the lyrics, "Ain't no sunshine when she's gone," to speak honestly instead.