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Please refer to the High-level Architecture Diagram to see the components of a ProGet installation.
The Database is a SQL Server database. The Web and Indexing Nodes are both .NET application that interact with the database using the standard SqlClient API. The connection string you specify in the configuration file is assigned to the ConnectionString property of that API.
All of this is very standard, but it's very likely that your DBA/Network Admins configured the SQL Server you wanted to use in such a way that a simple/intuitive connection string was not sufficient. Maybe they used a special port, security, etc.; they would be much better suited to answer your question about why it's so complicated :)
Also, please note that the ProGet Installer will create/update the tables and object in the database. ProGet itself does not modify (nor have the privileges to modify by default) the database structure. If you wish to install the database component outside of the installer, then you will need to follow the Manual Installation Instructions.