To complement Loïc Teyssier's excellent answer, this is the algebro-geometric interpretation of Stokes data, first in the case of $e^{1/x}$.
0. A zeroeth approximation to Stokes data in this case is the information that as $x\to 0$, $$e^{1/x}\ \longrightarrow \ \begin{cases} 0& \text{if }\text{arg}x\in (-\pi/2,\pi/2)\\ \infty & \text{if }\text{arg}x\in (\pi/2,3\pi/2) \end{cases}.$$ Here $x\to 0$ along rays (lines to the origin of constant argument). So Stokes data remembers how the limiting behavoir of the solution approaching the singular point depends on the argument.
Let's turn this into sheaf language. Take ODE on the disk $X=\Delta$ with singular point $0$, and local system of solutions $\mathscr{L}$ on $\Delta\setminus 0$. To be able to talk about the limiting behavior of solutions as $x\to 0 $ along rays, take the real oriented blowup at $0$ $$\pi \ :\ \widetilde{X}\ \longrightarrow \ X,$$ then $\pi^{-1}\mathscr{L}$ is a local system containing this information. Identify the fibre above $0$ with $S^1$. Write $\mathscr{V}$ for the restriction of $\mathscr{L}$ to $S^1$; this is where that information is stored.
1. A first approximation to Stokes data is a subsheaf $$\mathscr{V}^0\ \subseteq \ \mathscr{V}$$ given by the solutions with at worst a finite order pole in the given direction. Thus a germ $f$ lies in $\mathscr{V}^0_\theta$ if the size of $f(re^{i\theta})$ is bounded by $r^{-n}$ for some $n$ (this is not quite true, this needs to hold for a sector containing $\theta\in S^1$). In the $e^{1/x}$ example, this is $$\mathbf{C}_{(-\pi/2,\pi/2)}e^{1/x} \ \subseteq \ \mathbf{C}_{S^1}e^{1/x}.$$
2. The actual definition asks for (a little) more information about the limiting behavior. As a second approximation, the Stokes data is a collection of subsheaves $$\mathscr{V}^\alpha\ \subseteq\ \mathscr{V}$$ for every $\alpha\in \Omega^1_\Delta(\star 0)$ a meromorphic one form on $\Delta$ with poles only at $0$. A germ $f$ lies in $\mathscr{V}^\alpha_\theta$ iff $$f(re^{i\theta}) e^{-\int \alpha}$$ is bounded by $r^{-n}$ in a small sector containing $\theta$.
These subsheaves fit together to form a filtration, in that $$\mathscr{V}^\alpha_\theta\ \subseteq \ \mathscr{V}^\beta_\theta$$ whenever $e^{\int\alpha}e^{-\int \beta}$ has aforementioned boundedness property on a sector containing $\theta$. This gives a partial order on $\Omega^1(\star 0)_\theta$, for which the above is a filtration (a lie: you need to replace $\Omega^1(\star 0)$ by its quotient by the forms with at worst simple poles). Moreover, there's a grading on $\mathscr{V}_\theta$ for which this is the associated filtration.
$\infty$. A Stokes filtration is as in $2$ a filtration of $\mathscr{V}$ by a partially ordered sheaf, except you replace the Zariski fibre $\Omega^1(\star 0)_\theta$ with the etale fibre. In practice this means that you consider $\alpha=\sum_{n\ge n_0} a_n x^{n/k}dx$ for all $k\in\mathbf{N}$ and instead of just $k=1$.
So e.g. it contains information that $$e^{1/x}e^{\int \frac{dx}{\sqrt{x}^5}}\ =\ e^{1/x-2/3\sqrt{x}^3} \ \longrightarrow\ \begin{cases} 0&\text{if }\theta\in \pm(\pi,2\pi/3)\\ \infty&\text{if }\theta\in (-2\pi/3,2\pi/3) \end{cases}$$ where $\sqrt{x}$ is the positive square root defined off the negative reals.
In this language, Stokes lines are just the phenomenon that $f e^{-\int\alpha}$ flips between satisfying and not satisfying the boundedness condition for only finitely many angles $\theta$, so you can see the Stokes lines directly in the sheaves $\mathscr{V}^\alpha$.
Everything in this answer comes from
- La classification des connexions irrégulières à une variable, by Malgrange. http://www.numdam.org/item/CIF_1982__17__A1_0/
- Twisted wild character varieties, by Boalch and Yamakawa. https://arxiv.org/abs/1512.08091
The definition of a Stokes structure on a sheaf is $4.1$ of the first reference (it's the same as I've written above), how to give a Stokes structure in the ODE case is the top of page $7$. A Riemann Hilbert correspondence (which justifies the above definition of Stokes data) is theorem $4.2$.