There could be several reasons for your issue.
#1: Groove may not know where to look for your music. If this is the case, go into settings, and under Music on this PC, select Choose where we look for music. You can then add the folder locations of where your music library is located.
#2: It could be that the music is DRM (Digital Rights Management), and Groove is unable to verify proper ownership in order to play it.
#3: It could also be some reason that I'm unaware of, since I'm not an expert on the music side of things. Perhaps someone else will offer further insight.
If you are still having issues, you may want to try to download a different music player to play your offline music. There are several that are very good. I personally use VLC, and you can obtain it free from the Windows Store.