What's the point of duality?

For what it's worth, you're not the only one having trouble seeing the immediate relevance of dual spaces. In the preface to Michael Artin's algebra textbook, he says:

(2) The book is not intended for a "service course," so technical points should be presented only if they are needled in the book.

(3) All topics discussed should he important for the average mathematician.

[...] Sometimes the exercise of deferring material showed that it could be deferred forever -- that it was not essential. This happened with dual spaces and multilinear algebra, for example, which wound up on the floor as a consequence of the second principle.

When I read that as an undergraduate I thought "yeah, whatever" -- since what else could I do, not knowing what it was I was missing.

However, later when I came to differential geometry and tensor calculus (which I needed for general relativity) it turned out that duality is absolutely essential there. Then it wasn't very satisfying to lack the general algebraic grounding to fully appreciate what was happening. The books I were using did provide the bare essentials I needed to follow along, but it was also clear that there was a nice algebraic systematic hiding underneath all that which I didn't get to see all of. And it would certainly have been helpful to know that general theory before embarking on differential geometry.

Duality is a simple way to make new vector spaces. These dual spaces are useful in functional analysis, for example when you want to define the integral of a function, or you want to analyze a probability distribution. In this case there's a vector space of functions and a linear way to map those functions to numbers, which is natural to describe as an element of the dual space.

For finite-dimensional vector spaces, the dual is not so interesting because it looks like the original vector space. So it may not be very exciting in a standard undergraduate course. But for infinite dimensions, things are more interesting.

I have just been refreshing my memory from the Wikipedia article on Dual Space, which is a good summary.