Pure significance of line integrals of vector fields

In general, the vector line integral of a vector field $\mathbf F : \mathbb{R}^n \to \mathbb{R}^n$ along a smooth oriented curve $C$ is a measure of how much the field $\mathbf F$ "flows" along the curve $C$. That is, if $\int_C \mathbf F \cdot d\mathbf s > 0$, the field $\mathbf F$ has a net flow with $C$, while if $\int_C \mathbf F \cdot d\mathbf s < 0$, the field $\mathbf F$ has a net flow against $C$. This is a physical interpretation you can give to the line integral of a vector field.

We may perform line integrals of vector fields around closed loops to gather information about sources of those fields. This is true in electrostatics, where the line integral of the magnetic field about a closed loop produces an electric current within that loop. In electrodynamics, line integrals of electric and magnetic fields about a closed loop produces the time rate of change of magnetic and electric flux, respectively, through that loop.