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?