The whole idea of the VSTS DB is to get you set on the right path, ie. store database object definitions as .sql files, not as some fancy diagram. Any modification you do to the objects you do it by modifying the SQL definition. This way you get to do any modification to the objects, as permitted by the DDL syntax, as opposed to whatever the visual-designer-du-jour thinks you can and can't do. Not to mention the plethora of SQL code generation bugs associated with all designers out there.
The closes to a visual view is the Schema View, which shows tables, columns, indexes etc in a tree view and you can see the properties from there.
By focusing the development process and the Visual Studio project on the .sql source files, teams can cooperate on the database design using tried and tested source control methods (check-out/check-in, lock file, conflict detection and merge integration, branching etc).
the deliverable of a VSTS DB project is a the .dbschema file, which can be deployed on any server via the vsdbcmd tool. This is an intelligent deployment that does a a schema synchronization (merge of new object, modifies existing ones) and can detect and prevent data loss during deployment. By contrast, the 'classical' way of doing it (from VS Server eExplorer, or from SSMS) the deliverable was the MDF file itself, the database. This poses huge problems at deployment. The deployment of v1 is really smooth (just copy the MDF, done), but as soon as you want to release v1.1 you're stuck: you have a new MDF, but the production is running on its own MDF and does not want to replace it with yours, since it means data loss. Now you turn around and wish you have some sort of database schema version deployment story, and this is what VSTS DB does for you from day 0.