Hopefully this isn't too abstract, but for me it's when I have an idea for a story, I unintentionally build the plot around a theme. Not in a moral story kind of way--just in a "this is the thread that holds everything together kind of way." and then I kind of draw from things around me that have to do with that theme. So for Alchemist Burnouts, I had this theme of "why do you need to sacrifice so much of yourself in order to have worth and importance?" and while I don't think I ever need to answer that question (because that would make it a morality tale instead of expecting my readers to make their own conclusions) a lot of things around me suddenly started to click with it--books I've read, movies I've watched, and even things in history that I could echo in my work.
So in my writing I kind of think of inspiration in the same way I use reference when I draw art. It helps me get an idea of what works, what doesn't work, what's real, what's not real, and helps me to make a more realistic and personal connection when I write.