Your numbers live only in this browser, on this device. If you clear your browser data or lose the phone, they're gone unless you have a backup. To back up or move to another device: tap Copy backup, send the text to yourself, then on the other device paste it into the box and tap Restore from box.

Restoring replaces everything on this device with the backup. Your previous numbers are put in the box afterwards, in case you want to go back.