How can I show that these two integrals are equal?

You can separate them as follows

$$ \begin{align*} \iint e^{-x^2 - y^2} \mathrm{d}x \mathrm{d}y & = \iint e^{-x^2} e^{-y^2} \mathrm{d}x \mathrm{d}y\\ & = \int \left ( e^{-y^2} \int e^{-x^2} \mathrm{d}x \right ) \mathrm{d}y\\ &= \left ( \int e^{-x^2} \mathrm{d}x \right ) \left( \int e^{-y^2} \mathrm{d}y \right ) \end{align*} $$

Just in case, I will mention that whenever you have a double integral of the form

$$\int_{a}^{b} \int_{c}^{d} f(x) g(y) \mathrm{d}y \mathrm{d}x$$

you can separate it as a product of two integrals

$$ \int_{a}^{b} \int_{c}^{d} f(x) g(y) \mathrm{d}y \mathrm{d}x = \left ( \int _{a}^{b} f(x) \mathrm{d}x \right ) \left ( \int _{c}^{d} g(y) \mathrm{d}y \right ) $$

in the same way as before.