An interesting question. What does any field of knowledge teach you outside of its field?
Indulge an analogy for a moment. What does Mathematics teach you, outside of its domain? I think answers like 'it's connected to physics' aren't satisfactory; you haven't left the domain of mathematics yet! I think a better answer is that it teaches you a structured way of thinking. Mathematics requires you to formally define the things you know, develop a mathematical model of that situation, solve it using a set of formal rules, and then relate the result back to some useful real world situation.
Back to music theory. I think it's a great example of an imperfect analytical tool. We seek to describe, classify, reproduce, evaluate an art form that is subjective. We derive rules from existing works. We struggle to adapt those rules to some new style, that is undeniably music, yet doesn't fit our existing models. A significant subset of musicians know these rules (guidelines?) on a purely intuitive level, and in some cases, intentionally avoid knowing anything about them at all. Imagine if mathematics worked like that!
I'm sure there are other fields with similar characteristics; perhaps the study of paintings? I probably shouldn't display my ignorance of that field. But I think the mental gymnastics that we must perform to understand the tangled ball of rules, guidelines and exceptions that we've invented is a useful way of thinking. Humans are messy. Many (most) subjects are less formal, logical, even rational than something like maths. We need to know how to make sense of them, and I think the thinking cultivated by well taught music theory can be applied to these other areas.
That seems very philosophical. Possibly bad philosophy. As may be evident, this is also not an area I have much training in. But I decided to post this anyway. Feel free to disagree completely in the comments. I like learning new things.