Let $V$ be a vector space (over any field, but we can take it to be $\mathbb R$ if you like,
and for concreteness I will take the field to be $\mathbb R$ from now on;
everything is just as interesting in that case). Certainly one of the interesting concepts
in linear algebra is that of a *hyperplane* in $V$.

For example, if $V = \mathbb R^n$, then a hyperplane is just the solution set to an equation
of the form
$$a_1 x_1 + \cdots + a_n x_n = b,$$
for some $a_i$ not all zero and some $b$.
Recall that solving such equations (or simultaneous sets of such equations) is one
of the basic motivations for developing linear algebra.

Now remember that when a vector space is not given to you as $\mathbb R^n$,
it doesn't normally have a canonical basis, so we don't have a canonical way
to write its elements down via coordinates, and so we can't describe hyperplanes
by explicit equations like above. (Or better, we can, but only after choosing
coordinates, and this is not canonical.)

How can we canonically describe hyperplanes in $V$?

For this we need a conceptual interpretation of the above equation. And here linear
functionals come to the rescue. More precisely, the map

$$\begin{align*}
\ell: \mathbb{R}^n &\rightarrow \mathbb{R} \\
(x_1,\ldots,x_n) &\mapsto a_1 x_1 + \cdots + a_n x_n
\end{align*}$$

is a linear functional on $\mathbb R^n$, and so the above equation for the
hyperplane can be written as
$$\ell(v) = b,$$
where $v = (x_1,\ldots,x_n).$

More generally, if $V$ is any vector space, and $\ell: V \to \mathbb R$ is any
non-zero linear functional (i.e. non-zero element of the dual space), then
for any $b \in \mathbb R,$ the set

$$\{v \, | \, \ell(v) = b\}$$

is a hyperplane in $V$, and all hyperplanes in $V$ arise this way.

So this gives a reasonable justification for introducing the elements of the dual
space to $V$; they generalize the notion of linear equation in several variables
from the case of $\mathbb R^n$ to the case of an arbitrary vector space.

Now you might ask: why do we make them a vector space themselves? Why do we want
to add them to one another, or multiply them by scalars?

There are lots of reasons for this; here is one: Remember how important it is,
when you solve systems of linear equations, to add equations together, or
to multiply them by scalars (here I am referring to all the steps you typically
make when performing Gaussian elimination on a collection of simultaneous linear
equations)? Well, under the dictionary above between linear equations
and linear functionals, these processes correspond precisely to adding together
linear functionals, or multiplying them by scalars. If you ponder this for a bit,
you can hopefully convince yourself that making the set of linear
functionals a vector space is a pretty natural thing to do.

Summary: just as concrete vectors $(x_1,\ldots,x_n) \in \mathbb R^n$ are naturally
generalized to elements of vector spaces, concrete linear expressions
$a_1 x_1 + \ldots + a_n x_n$ in $x_1,\ldots, x_n$ are naturally generalized to linear functionals.