Understanding the definition of an abstract manifold
The map $F_1$ is a map from $U_1$ to $V_1$. And $F_2^{\,-1}$ is a map from $V_2$ to $U_2$. But $F_2^{\,-1}\circ F_1$ is not defined on the whole $U_1$; $(F_2^{\,-1}\circ F_1)(x)$ makes sense only for those $x\in U_1$ such that $F_1(x)$ belongs to the domain of $F_2^{\,-2}$. And this happens exactly when $x\in F_1^{\,-1}(V_1\cap V_2)$, that is, when $x$ is such that $F_1(x)\in V_1\cap V_2$.