Have you selected anything to sync?
Click on the iPhone button at top right to select the iPhone.
You should now see a new bar with the name of your phone, "Summary Info Apps Music Movies" through "Photos", maybe the device name again, and a blue "Done" button.
Click on "Music" to activate the music pane.
You should now be able to select which of your songs, albums, artists, and whatnot you want to copy ("sync") onto the phone. (If you have iTunes Match active, you won't get that choice. I do, so I can't quote the exact text.)
If you also want to sync movies, TV shows, podcasts, or whatever, repeat using those buttons.
Once you have something selected to sync, clicking the "Sync" button at the lower right should do something.