How do I fix my Game Center as it won't open?

Can usually be due to storage being too full. Check you have free space. You can try a reset all settings. Settings > General > Reset > Reset all settings. This will change your background, passcode, network connections all back to default. If that does not resolve it you'll have to restore it on iTunes. Back it up first else you loose everything. This will fix it for sure if you set it up as new buck restoring backup should be okay if you need your data back.