I've been trying to explain this to my parents for quite some time- and since I'm primarily interested in syntax and semantics, with little interest in applied linguistics, it has not been an easy task.
I think that the closest I've come to explaining it is that language is not only something that every single person in the world cares about and has opinions on (just ask a group of people at a party how they feel about twerk being in the dictionary) AND something that every human uses nearly every day BUT, I think more importantly, has a structure that's hidden from the majority of its users. Until you start to learn about linguistics, you have no idea that there is motivation behind how your brain allows you to construct your own language. You have no idea that those weird neologisms and derivations have rules and histories behind them. You have no idea why colorless green ideas can sleep furiously but you can't tell me what that means.
Linguistics is an interesting science because, when it comes to grammaticality judgments, we already have all of the data at our fingertips (literally if you study sign language!...sorry I couldn't resist). And yet the fact that there are still so many questions is a testament to what a perfectly tempting puzzle language is. And that clearly matters to some people.