The designer tracks changes on the meta-data when created by the user or mapping code and will emit the changes into DDL SQL to modify an existing schema.
So if you rename a field in a table, it will create an ALTER TABLE statement and drop the column and re-add it, if there's no proper way to do this in the SQL of the target database. SQLServer has sp_rename, so a rename will use that stored proc instead
For types etc, as long as there's an ALTER statement, DROP/ADD or stored proc, we're using it, so changing a type in a column uses an ALTER COLUMN statement, however this can still require you to step in and add migration code to convert the data first.
The changes are emitted into change scripts, and as only the changes are emitted (or you can opt for a complete new schema) it's not hard to rework the changes into a change script which migrates your schema WITH data.
We won't emit data migration code, as that's impossible in a lot of cases: migration of data often requires special code which is written by the DBA and migrations of databases with data require migration tests on copies of the live data before it's ran on the database. For example data which requires a function call to be migrated properly due to a type change on a column requires custom code we can't provide for you. So in that light it's not that odd that red gate says it's not really possible.
So, in many cases, the changes to the meta-data are performed through ALTER statements, DROP/ADD statements, proc calls and data is migrated by the RDBMS. In other cases, you need to write some extra SQL, however it's not that hard to do as only changes are emitted to the change script so in general it's not that much SQL. Example, if you rename a field, add a field and change a field's type, you get 3 ALTER COLUMN / ALTER TABLE statements.
Of course, the DDL sql is generated using the code generation pipeline with its own preset and templates, ran on the project model, so you can alter the templates, add your own code if you want to or generate different code than DDL SQL.
The goal is that a developer can manage the database schema from the designer, by modifying the entity model.