You can build a bit of mathematical intuition for the concept now that you get the basic idea. Try and work out from the functor laws why functors preserve shape. The laws are very straightforward:

    map id c = id c

(Mapping the identity function is the same as simply applying the identity function -- or with a little more category theory, the functor maps the identity function in the base category to the identity function in the functor's category)

    map f (map g c) = map (f . g) c

The second law is also simple -- mapping one function over the container then mapping another function is exactly the same as mapping the composition of the functions over the container. This one is the basis of stream fusion, a very important optimization, that allows you take two traversals of a container and turn them automatically into just one traversal.

The preservation of structure follows from just these two laws and parametricity (The element type of the containers is generic and therefore unknown, this greatly restricts what operations are available on the elements of the container. You can't, for example, conjure up a new value of the element type to insert.). I strongly recommend trying to figure out how.