This is kind of a big squooshy question (or series of questions), which I will try to cast in a more precise form. Apologies if I don't succeed.
Context: I'm an amateur set theory/category theory enthusiast with special interest in stratified set theories. My knowledge is substantial in odd spots, but not very well rounded. Because of the ubiquity of talk of models in the literature, it seemed like a thing I should learn, but I'm having a difficult time because I feel like I don't grasp certain informal aspects of "model theory in practice", even though I technically understand about the first half of David Marker's "Model Theory: An Introduction", for instance.
Questions: What are the big questions one wants to answer with models that would either be unapproachable or extremely painful without it? I gather questions of consistency are important ones, but are there other main ones? When does one want to know about the second order properties of elementarily equivalent models, and why? Are there even general answers to these questions, or am I asking things I would have to ask, say, another set theorist about to get anything informative?
I hope this isn't too poor a question; I'm flailing a bit to articulate beyond "it's cool, but WHY???"