I just found this thread online that answers why, but it doesn't really answer how. I have almost nothing connected to my iCloud and a lot of my library isn't available on Apple Music, so I'm not quite sure what to do.
If iTunes can't recognize the song, it will upload your copy to iCloud and let you stream that version to your computer and iOS devices. (These songs don't count against your storage; they're included in the price of Apple Music.
Mine did not do this.
When you activate iCML on your computer and your iDevice, you can no longer sync your music to it on iTunes!
Instead, it just gets access to your whole library through iCML and you can download whatever songs you want offline over WiFi or cellular.
This is a bit ridiculous, in my opinion. Why should I not be able to pay for a streaming service AND have my own personal music collection at the same time?