Suppose you are studying some finite dimensional topological vector space $X$ (which, in turn, must be isomorphic to $\mathbb{C}^d$), then I guess the most natural thing to do is to introduce coordinates. Or, equivalently, choose a basis $\{e_j\}_{j=1}^d$, so that each $x=\sum x_j e_j\in X$ is represented by $(x_1,x_2,\dots, x_d)$. The mappings $x\mapsto x_j$ and their linear spans are exactly continuous linear functionals for $X$, and for $X=\mathbb{C}^d$,the dual space is also $\mathbb{C}^d$.So you can see the dual space for infinite dimensional spaces is just a generalization of coordinates.
However, continuous linear functionals are much more useful than coordinates, mainly because the topological structure and algebraic structure are not so well-behaved for infinite dimensional spaces (they determine each other for finite dimensional spaces). They become indispensable throught the collection of theorems bearing the name of Hahn-Banach, with which (at least in norm space setting) one can separate points from closed subspaces, from convex bodies, etc.
So I think from here it is already clear continuous linear functionals are like coordinates, but they can somehow blend both algebraic (linear) structure and the topology. When the space is complicated, study the functions over the space (that respect certain properties of the space). This is like the theme behind dual spaces.
From another point of view, they are also very natural in the sense that so many objects (evaluation, integration and distribution) can be realised as continuous over certain spaces.