Later you can restore the saved data (except MMS) back to the phone, especially useful when switching to a new device.
IMAP access needs to be manually enabled in Gmail, see the website and FAQ for more information. You can of course use your own IMAP server for backups, giving you full control over your data.
SMS Backup+ is a free open-source project which has been in active development since the early days of Android, completely ad and tracking-free, supported through voluntary donations.
Many thanks to the numerous people who have donated, contributed feedback, code or translations – detailed credits can be found in the app or on the website.