Hi, DannyBaek!
Thanks for using the Apple Support Communities. Based on your post, I understand you're trying to listen to songs via Apple Music, but are prompted to sign in first, though you're pretty sure you are. I love Apple Music and listen to it daily, so I want to make sure you can access your songs!
First, just to be sure, I would recommend verifying you're signed into the iTunes Store with the Apple ID you set your Apple Music membership up with. You can do this by clicking Account in the iTunes menu bar. If you are signed in, your name and Apple ID will be listed at the top of the drop down menu. If it isn't the right Apple ID, click Sign Out from the menu, and then sign in again using the right one. Instructions on exactly how to do that can be found in the article below.
iTunes 12 for Windows: Manage your iTunes Store account
Once you're sure the correct Apple ID is signed into the iTunes Store, make sure iCloud Music Library is turned on under Edit > Preferences, as shown in this screen shot:
Once you've confirmed you are signed in with the right Apple ID and iCloud Music Library is enabled, if you're still not able to play your music, make sure your internet connection is working and check for any iTunes updates. If you're not sure how to check for updates, the article below will help.
Get the latest version of iTunes
Then, close iTunes and restart your computer. When the computer is on again, open iTunes and try to play your music.
If you still have trouble playing your music after restarting, I recommend completely uninstalling iTunes and all of its components following the exact steps in the article below, then redownload a fresh copy from this site: Download iTunes
Remove and reinstall iTunes and related software components for Windows 7 and later
Happy listening!