Consider what happens if you take a $D$-module on an algebraic curve (with field of fractions $K$) and remove all the information on the singularities. You can achieve this by tensoring over the structure sheaf with $K$, obtaining a module for the ring of differential operators on $K$. This ring is generated over $K$ by differentiation along a single meromorphic vector field (since it's generated by differentiation along all vector fields. So a module over it is just a $K$-vector space with a semilinear action of this differentiation. This will usually be finite-dimensional (I think always for holonomic $D$-modules).

To pass to the Galois theory, we pick a specific vector field and view it as a derivation $D$ on $K$, so we have a finite-dimensional vector space with an action of $D$.

From a finite-dimensional vector space with an action of $D$ one can make a differential field extension using Picard-Vessiot theory. Take a ring generated by independent transcendentals corresponding to  basis of this vector space, with the $D$ action given by the $D$ action on the vector space, mod out by a maximal differential ideal, and take the field of fractions.

Any field extension generated by solutions of ODEs arises this way, because we can construct from an order $n$ ODE the vector space generated by a formal solution and its first $n-1$ derivatives and take the corresponding ring, which maps to the field, and the kernel is a differential ideal.