In addition to its literal sense that is interpreted by giving ("delivering") and rope ("rope"), there is one that is almost literal, and has to do with mechanical gadgets that to move used a spring that was tightened by pulling a rope, and when recovering its shape slowly by a clockwork mechanism animated toys or small machines, and it is then said to "give [the pull to the] rope" for "making them work". From there appears the meaning of "bait, encourage, promote to do something". See "having rope for a while".
