Apple ID login

A few ideas:

1. Are you sure you have an internet connection (yes, I know, it's obvious but I had to ask.) Try to google something to make sure.
2. Open Safari, go to iCloud.com and see if you can log in
3. Restart your phone

Some things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Backup - Restore your device from an iCloud or iTunes backup.