My textbook has the following example:
Let $f,g:\mathbb{R}^2\to\mathbb{R}$ be $C^1$ functions such that $f(x+k,y+l)=f(x,y)$ and $g(x+k,y+l)=g(x,y)$ for $x,y\in\mathbb{R}$ and $k,l\in\mathbb{Z}$. Then the differential equation in $\mathbb{R}^2$ given by:
\begin{cases} x'=f(x,y)\\ y'=g(x,y) \end{cases}
can be seen as a differential equation on the torus $\mathbb{T}^2$. Clearly, the above equations have a unique solution (that is global, i.e., defined for all $t\in\mathbb{R}$ since the torus is compact).
It's this last sentence that I don't understand. Is this a widely known theorem in ODEs? The author doesn't discuss this in prior sections.
That follows essentially from the continuous dependence of ODE with respect to the initial condition. Let $x_0 , y_0 \in \mathbb T^2$ and $(x(t), y(t)) : [0, \epsilon) \to \mathbb T^2$ be the unique solution to the ODE $$\begin{cases} x' = f(x, y) \\ y' = g(x, y).\end{cases}$$ Then there is an open set $U$ in $\mathbb T^2$ containing $x_0, y_0$ so that the ODE can be solved for $T = \epsilon/2$. By compactness of $\mathbb T^2$, there is $\delta >0$ so that the ODE can be solved for $T=\delta$ for all initial conditions $(x, y) \in \mathbb T^2$. Then it can be solved for all $t\in \mathbb R$, see here.