The error says:
Failed to create extension manager for the target platform 'Microsoft.Data.Tools.Schema.Sql.Sql120DatabaseSchemaProvider'.
My teammate created the DB Project, but when I pulled it down from source control it failed to create.
The solution is to get an update to the "SQL Server Data Tools". You apparently have an older version.
Look in Visual Studio 2013 under Tools-->Extentions and Updates. On the left pane from there, click on Updates. Hopefully you'll be able to find the update you need there.
(Alternatively, run a repair installation on "SQL Server Data Tools" under Windows' Control Panel-->Programs.)