How a theory can be categorical of a large cardinality?

If all the types are principal, then you cannot omit them.

Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.

There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.

Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $\aleph_0$-categorical.

There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $\aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $\mathbb{Q}$-vector spaces.

When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{i\in I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.

This might seem like a problem if $T$ is $\kappa$-categorical for some uncountable $\kappa$, since given any uncountable set of types, we can realize them all in a model of size $\kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $\kappa$.

But all we've done is proven that $T$ is $\omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.