Universes in dependent type theory
The question under what circumstances we need to jump from a universe to one higher in the hierarchy is a good one. Having the hierarchy and the ability to climb it is important. You need to jump levels when you want to treat a universe as a type or as part of a type. For example to define functions of (non-dependent) type
$$
   A \rightarrow \mathcal{U}_i
$$
you must show that $A \rightarrow \mathcal{U}_i$ is in a universe.
But that cannot be  $\mathcal{U}_i$ or some smaller universe. So what do we do? To deal with the problem (without using the unsound $\mathcal{U}_i : \mathcal{U}_i$), we need to jump up a universe. The rule that
enables us to make this jump is $\mathcal{U}$-Intro
$$
   \frac{\vdash \Gamma : ctx}{\Gamma \vdash \mathcal{U}_i : \mathcal{U}_{i+1}},
$$
given in Appendix A.2.3. The very point of the hierarchy of universes is that we can do this. This can be seen as a safe approximation of having universes contain themselves.