See Data backup overview. Since changelog is takingtalking of Android 6 compatibility, it is referring to Android auto backup (see table).
Enabled by default. Apps can opt out by disabling backups.
Contextually, it implies Textra had opted out of back up earlier and now has it enabled.
There is no option for the user to choose which apps can be backed up or not. Either they backup or not (search settings for backup to see the toggle switch). If backup is enabled, apps that are enabled for backup are backed up.
Developers can choose to exclude some parts of app data even if backup is enabled. Backup data size is limited to 25 MbMB and doesn't count for your drive storage usage. Users have no control on what is backed up, so if you don't trust Google (good idea) look for custom ROM options as suggested by Izzy in comments or turn off backup in settings.