I would seriously question who you take advice from
Seriously though, its essential to bring your weight through in delivery. The falling away aspect you mention is unrelated and a symptom of another problem.
Its the same when you drive the ball when batting. You do not hit the ball by swinging your arms through the line but get your power from leaning into the ball and bringing your weight forward into the ball.
Imagine a ruler being bent back. The further you bend it the more violently it snaps back. Its the same with fast bowling. The more you 'snap' the weight from back to forward the more momentum you can put in the ball.
Apologies to those with dial-up, but just to prove a point
Gough, Weight back ready to explode forward
Botham, Weight back ready to explode forward
Imran, Weight back ready to explode forward
Shoaib, Weight back ready to explode forward
Lee, Weight back ready to explode forward
Dilley, Weight back ready to explode forward
Weight transfer, from back to forward, in the coil and delivery is essential to bowl quick.