Feeling our way into a new formal system is part of our (messy, informal) pebblecraft. Sometimes people propose formal systems starting with their intended semantics (roughly, model theory). Sometimes people propose formal systems starting with introduction and elimination rules (roughly, proof theory). If the latter, people sometimes look for a semantics to go along with the syntax (and vice versa, of course).

For example, lambda calculus started with rules for performing beta reduction. In talking about lambda calculus, people refer to it as "functions from functions to functions". Mathematicians knew that there was no nontrivial set S with a bijection to the set of all functions from S to S. So something else must be going on. Dana Scott invented domain theory partially to solve this problem. Domains have additional structure, such that some domains D do have bijections with the set of all structure-preserving maps from D to D. http://en.wikipedia.org/wiki/Domain_theory Similarly, modal logics started with just proof theory, and then Saul Kripke invented a semantics for them. http://en.wikipedia.org/wiki/Kripke_semantics

There's always a syntactical model which you can construct mechanically from the proof theory (at least, that's my understanding of the downward Lowenheim-Skolem argument) - but Scott and Kripke were not satisfied with that syntactical model, and went looking for something else, more insightful and perspicuous. Adding a "semantic" understanding can increase our (informal) confidence in a formal system - even a formal system that we were already working with. I'm not sure why it helps, but I think it's part of our pebblecraft that it does help.

Perhaps adding a "semantic" understanding is like another bridge between informal and formal reasoning. These bridges are only partly formal - they're also partly informal, the concepts and gesturing around the equations and proofs. Having one bridge is sufficient to go onto the Island of Formality, do some stuff, and come off again, but might be more convenient to have two, or three.