Consider
i) Find the vector fields $V_1, V_2, V_3$ which generate the following smooth one-parameter groups of transformations of $\mathbb{R}$ : $$ x \mapsto \psi_1^s x=x+s, \quad x \mapsto \psi_2^s x=e^s x, \quad x \mapsto \psi_3^s x=\frac{x}{1-s x} . $$ ii) Deduce that these vector fields generate a group of transformations of the form $$ x \mapsto \frac{a x+b}{c x+d}, \quad a d-b c=1 . $$
I don't understand what is meant by a group of transformations generated by the vector fields. I don't understand how we generate elements from these. Could someone elaborate on this?
Hint: "Generate" in this context means to "generate infinitesimally". So one (complete) vector field generates an action of $\mathbb{R}$ by diffeomorphisms (by the Existence and Uniqueness Theorem in ODEs; see e.g. What is the relationship between the vector fields of conjugate flows?), and more generally a Lie algebra $\mathfrak{g}$ generates an action of a Lie group $G$ by diffeomorphisms. Standard Lie theory then guarantees that, there is up to isomorphism a unique such $G$ (assuming connectedness).
Reverse engineering the second part, you would need to verify that the smallest Lie subalgebra of the Lie algebra of $C^\infty$ vector fields on the real line containing $V_1, V_2, V_3$ is isomorphic to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ of $2\times 2$ traceless matrices.
It might also be useful/exciting to note that $\psi_2$ is analogous to the geodesic flow, $\psi_1$ is analogous to the unstable horocycle flow and $\psi_3$ is analogous to the stable horocycle flow (on the unit tangent bundle of a surface of constant negative curvature).