Why don't songs in iTunes appear in Music on the iPhone?
If you want your music to be uploaded to iCloud, you must sign up for iTunes Match.

Make sure you're running the latest version of OS X and iOS. (El Capitan - iOS 9 are the stable releases right now, although you can get the public beta of macOS Sierra and iOS 10).

Then make sure the Apple ID you've entered in the Music App on iOS is the exact one on the Mac's iTunes. If it didn't work, make sure both devices have been connected to the internet since setting the Apple ID.

By the way, "Songs of Innocence" is an album by Irish rock band U2, which was a 'gift' from Apple to you. If you want it removed, see this.

Of course, connecting your iOS device to your Mac can help you sync them. This webpage has got a table which lets you know how to do deal with some of these similar problems.