Why marginal revenue curve is twice as steep as demand curve in various market structure?

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The calculus-free answer

Think of the affect incremental increases in quantity has on total revenue. If you make a simple graph with a demand curve and draw boxes representing total revenue examine how the total area of the box (representing the total revenue) varies as quantity increases. With a linear demand curve, as you move down the curve the box becomes larger and larger in area until you reach the curve's midpoint. This means that the MR up to this point was positive because TR was increasing. After this point the area of the box declines, this means that from this point forward the MR is negative because TR is decreasing. This is why the MR curve hits zero dollars at the half the quantity that the demand curve hits zero dollars. Hope this helps.
-DVE

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