The fractional Sobolev spaces are important when formulating elliptic boundary value problems in Sobolev spaces.
Consider the Dirichlet problem for the Poisson equation. That is, you are interested in a solution $u$ to $\Delta u = g$ on a bounded open domain $\Omega \subset \mathbb{R}^n$ that satisfies a given boundary condition $u | \partial \Omega = f$ for some $f$ that is defined on $\partial{\Omega}$.
How do you formulate "boundary" conditions when functions in Sobolev spaces aren't necessarily even continuous? The boundary $\partial \Omega$ of (a sufficiently nice open subset) $\Omega$ is of measure zero and functions in $H^k(\Omega)$ are defined a priori only a.e, and changing them on a measure zero subset doesn't affect them.
Still, if $\partial \Omega$ is nice enough (say, a $C^k$ manifold), and if $k \geq 1$ is an integer, one can show that there are trace operators $T : H^k(\Omega) \rightarrow L^2(\partial \Omega)$, that extend continuously the usual restriction map $f \mapsto f| \partial \Omega$ on $C^{\infty}(\Omega) \cap H^k(\Omega)$. What is the image of $T$? That is, what are all the possible "boundary values" of a function in $H^k(\Omega)$? It turns out to be precisely $H^{k-\frac{1}{2}}(\partial \Omega)$!
Using this, one can formulate and prove uniqueness and existence results for solutions of PDEs in Sobolev spaces. For example, one has that the map $ u \mapsto (\Delta u, u|_{\partial \Omega}) = (\Delta u, Tu) $ is an isomorphism of $H^1(\Omega) \rightarrow H^{-1}(\Omega) \times H^{\frac{1}{2}}(\Omega)$, and so, given $g \in H^{-1}(\Omega)$ and $f \in H^{\frac{1}{2}}(\Omega)$, the Dirichlet problem for the Poisson equation $\Delta u = g$, $u|_{\partial \Omega} = f$ has a unique solution in $H^1(\Omega)$.
There are also higher order trace operators, which correspond to normal derivatives of various orders, whose image again lies in fractional Sobolev spaces.