How is a solid doughnut, a disk of circles? I understand how its a circle of disks but not how its a disc of circles. Solid doughnut = $D^2$ X $S^1$. Animation would elate me.

Not an animation, but I hope this gives you some of the idea.

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For each point on the disc, we have a circle. Taking the union of all the circles together gives us the solid torus.

In general, if $X$ and $Y$ are spaces, then their product $X\times Y$ is naturally homeomorphic to the product $Y\times X$ in the other direction.