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Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.

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It's basically this: When you do single variable integration, you are fitting rectangles under a curve and letting the width of the rectangles get smaller and smaller. This will give you an approximation of the area under the curve.

When you do double integration over a region, you fit boxes under a surface. Just as in the single variable case, as you let the area of the base of these boxes get smaller and smaller, the volume of all the boxes will get closer and closer to the volume under the surface.

This website seems to have some pictures of this:

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