What is meant with unique smallest/largest topology?

Solution 1:

If a topology is largest, it's unique automatically. Suppose $\tau_0$ and $\tau_1$ were both largest topologies. Then $\tau_0\supseteq\tau_1$ and $\tau_1\supseteq\tau_0$, hence $\tau_0 = \tau_1$.

Maximal, on the contrary would not have to be unique.

Solution 2:

The set of all topologies contained in $\{T_{\alpha}\}$ has a partial ordering so it has a sense of maximal element. You want to show that if $\tau$ is maximal with respect to this partial ordering then $\tau \subset \tau_0$ where $\tau_0$ is the topology you mentioned. Since $\tau$ was maximal you will have $\tau=\tau_0$.