Why can’t I sign back into my App Store with my Apple ID after signing out?

Maybe this read will help: If you can't connect to the App Store, iTunes Store, or other Apple services https://support.apple.com/en-us/HT201400

What happens when you try? Error message(s)?

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.