Can I not get downloaded files from iTunes off my computer (Mac) and on another device (Windows) for DJ-ing?

I'm getting married soon, and I'm trying to get music to my wedding DJ before the wedding. He owns a windows computer and shuffles all his music from locally stored MP3 files on a hard drive. He doesn't have one song, so I bought it from iTunes through Apple Music. Now I've bought it, but there is no file anywhere and no option to download onto my hard drive. What gives? Why is there even an option to buy music if you can't own it? Please don't tell me I need to buy it again from another online source.