Proof:

It is an exercise that the space obtained by collapsing the boundary of $I^2$ to a singleton is homeomorphic to $S^2$. Let $\eta_1$ denote the quotient map $I^2\longrightarrow S^2$ which collapses the boundary to a singleton and likewise let $\eta_2:S^1\times S^1\longrightarrow (S^1\times S^1)/(S^1\vee S^1)$ be the quotient map. The map $\phi : S^1\times S^1\longrightarrow S^2$ given by the prescription

$$\phi(\exp(2\pi ix), \exp(2\pi iy)) = \eta_1(x, y)$$

is well-defined and surjective. Since $\eta_1$ is continuous it follows that $\phi$ is continuous (why?). There is a unique bijective map ${\overline \phi}:(S^1\times S^1)/(S^1\vee S^1) \longrightarrow S^2$ such that $\overline{\phi}\circ \eta_2 = \phi,$ from which follows that ${\overline \phi}$ is continuous and a closed map since the domain is compact and the codomain is Hausdorff. Hence ${\overline \phi}$ is a homeomorphism between $(S^1\times S^1)/(S^1\vee S^1)$ and $S^2$.