What will Amazon RDS manage on the behalf?
Depends on which service in RDS use choose. If you use Aurora Serverless, you don’t really have to manage anything. If you go with standard Aurora, MySQL, PostgreSQL, etc., you will have to determine how many read-replicas you need, choose instance sizes, etc.
But will all of these services, things like automated backups, service patches, and server maintenance are all taken care of for you.