OK I think I understand the question and I think I am going to give you the correct answer or something close to it.

Control cards put out low voltage signal to that coil, the resistance slows the voltage passing through it and allows it to build to whatever the voltage range is for the coil to pull that contactor closed.

Simplest example I can think of is a garden hose allowed to flow freely only allows water to come out of the open end a foot or two. Now put a spray nozzle on the end of the hose. That now puts resistance and makes that water flow through a smaller orafice and now the water will spray twenty feet.

I hope I explained that correctly, its been about twenty years since I went to basic electric school. If I didnt hopefully someone will correct me.

Oh and the increase in resistance from end to end to coiled together is just surface contact. The more wire touching itself just concentrates the resistance.