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I'm just starting to go deeper in my linear algebra education, and this question: Why do we care about dual spaces? helped my intuition and motivation immensely, especially Matt E's answer.

So, I'm curious if that kind of explanation can be furthered to transposes/duals of linear maps? (Or of course, any kind of explanation that could enlighten me!)

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If you seek elegant and powerful examples of applications of linear functionals, duality, adjoints etc. then all that and more can be found in Umbral Calculus. You can find a nice introduction in Steven Roman's book The Umbral Calculus. These few fundamental ideas serve to explain, unify and organize many hundreds of diverse properties of special functions that arise in combinatorics.

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