Peano arithmetic, as in classical first-order logic with the peano axioms is a small subsystem of MLTT with natural numbers and no universes. If you come from a set theory background, then universes are really akin to Grothendieck universes, or large cardinal axioms. You start out with a powerful theory and then improve it by repeatedly adding statements of the form "and the theory so far is consistent", by giving an internal model of "the theory so far".

There are type theories with universe variables, which essentially allow you to add an arbitrary (but finite) number of additional universes. The rules for this are straightforward, even though a consistency proof is of course only possible relative to type theory or set theory with more universes...

In set theory, one typically adds an axiom scheme that states something like "There is a Grothendieck universe containing this set". Iterating this gives you a similar tower of universes.

There are a lot more constructions that go beyond this and the funny thing is that as far as anyone knows they are all consistent. For instance, type theories with induction-recursion allow you to generate internal universes closed under certain operations while staying in the same universe. So one universe with induction-recursion allows you to show MLTT with an arbitrary finite number of universes consistent.