Denote by $\mathfrak{M}([0,1])$ the $\sigma$-algebra of Lebesgue measurable functions. The space $S([0,1])=\operatorname{span}\{1_{A\times B}:A\in\mathfrak{M}([0,1]),B\in\mathfrak{M}([0,1])\}$ is dense in $L^2([0,1]\times[0,1])$ Since $\{f_n(y)\}_{n=1}^\infty$, $\{g_n(x)\}_{n=1}^\infty$ are basis of $L^2([0,1])$ you can approximate each function in $S([0,1])$ by linear combiantions of functions $f_n(y)g_k(x)$. So $ \operatorname{Closure}\left(\operatorname{span}\{f_n(y)g_k(x)\}_{n=1,k=1}^{\infty,\infty}\right)=\operatorname{Closure}(S[0,1])=L^2([0,1]\times [0,1]) $