Explanation for example of flow generated by vector field
The components of the vector field are $X_1(x, y) = -y$ and $X_2(x, y) = x$. These components define a system of ODEs for a curve: $$ \dot{\gamma}(t) = X(\gamma(t)), $$ or more explicitely $$ \begin{align} \dot{\gamma_1}(t) &= X_1(\gamma(t)) = -\gamma_2(t), \\ \dot{\gamma_2}(t) &= X_2(\gamma(t)) = \gamma_1(t). \end{align}$$ The solution $\gamma(t)$ is the flow generated by the vector field. The function $\sigma$ solves the given system of ODES, therefore it is a flow. For further information take a look at this script from the ETH Zurich