Let the distance between opposite corners be $c$. First we need to consider the length of the diagonal of a face of the cube. Let’s call this length $b$. $b$ is the hypotenuse of a right-angled triangle with both other sides of length $a$: Hence, $b^2=a^2+a^2=2a^2$. $c$ is the hypotenuse of another right-angled triangle with the other two sides of lengths $a$ and $b$: Therefore \begin{align*} c^2&=a^2+b^2\\ c&=\sqrt{a^2+2a^2}\\ &=\sqrt{3}a \end{align*} So the distance is $\sqrt{3}a$.