This is one of those things that people said in the 1950s
, people with greased hair
and collared shirts.
To "throw someone the connection" means to clear things up, to elucidate one's motives.
Syntax (in tones of great sarcasm): "Well, Frank, you still don't get it, do you? I'll just have to throw you the connection."
Possible origin: This sounds like a reference to telephone switchboards to me, but I could be wrong. Anybody else know the answer?