Now it is clear to see how ODEs can define a tangent vector field on a manifold.

For this, we use first-order, explicit ODEs in \$$\mathbb{R^n}\$$, which give the tangent vector in \$$\mathbb{R^n}\$$ as a function of the coordinate point \$$C(s) \in \mathbb{R^n}\$$.

Then just send those tangent vectors in \$$\mathbb{R^n}\$$ through the aforementioned vector-space isomorphisms, i.e. through the charts, to get the tangent vectors for the manifold proper.