Why is this the basis for the indiscrete topology?
If $\mathcal{B}$ is a basis, then for each open set $U$ there is a subset $\mathcal{A}\subseteq\mathcal{B}$ such that $U$ is the union of all of the elements of $\mathcal{A}$. In this example, for $U=\emptyset$, you can just take $\mathcal{A}=\emptyset$. The union of all the elements of $\mathcal{A}$ is the union of no sets, which is just the empty set.
By the way, you should say a basis, not the basis, since it's not the only basis for this topology: $\{\emptyset,X\}$ is also a basis.