One often thinks of a homomorphism as something that "preserves the structure" of an object, but it is often better to think of it as something that "does not add new information to the object".

The most basic example is in $\mathbf{Set}$. The "information" of a set is its cardinality. The defining feature of the morphisms here -- functions -- is that they do not "create new cardinality". A point cannot be mapped to multiple points.

Similarly in $\mathbf{Top}$, the "information" of a topological space is the distinguishability of two points. This notion of distinguishability "includes" those used in the separation axioms (so e.g. a continuous map cannot take you from the indiscrete space to the discrete space), but is more general and vague -- the general idea is that two things touching make them "kinda indistinguishable", so you can't tear them apart.

This idea clearly has to do with inverse images -- we're saying that for things in the codomain, the information they carry must have already existed in their preimage. In fact in the previous example, the way to formalise this notion of being "kinda indistinguishable" is best formalised in the language of open sets, and a continuous map can't create new open sets.

Perhaps the clearest example comes from the category of measurable spaces $\mathbf{Prob}$. Here, the sigma fields really do represent information, and the definition of a measurable function (or random variable) is that it cannot talk about things that can't be measured. I.e. if a piece of apparatus just measures the number of heads in a coin-flip experiment, we can't have a random variable asking if the first toss was a head. Once again, this notion of "not adding new information" directly corresponds to preimages.

A bit more detail in my post here.