The first step to moving a SQL Azure Database to a local server is getting access to a backup of the database. The easiest way is to export the database to a .bacpac file directly from SQL Server Management Studio (SSMS). This .bacpac can then be imported, still using SSMS, to a local database. In order for this to work you need to have SSMS for SQL Server 2014 or later (download the lastest version here). It is also possible to use CU6 for SQL Server 2012.
In SSMS, right click your SQL Azure database and select "Tasks" then "Export Data-tier Application...". Follow the wizard to export the database to a .bacpac file.
With the .bacpac saved locally, the next step in restoring a Azure SQL Database is to right-click on the Database container in SQL Server Management Studio. From this context menu, select the option to "Import Data-tier Application..." and follow the wizard to import your database.
Sometimes the import operation fails for various reasons. See below for possible workarounds.
The import operation fails with an out-of-memory exception
When this happens it is usually possible to use the SqlPackage.exe command line tool to import the .bacpac.
SqlPackage.exe /a:Import /sf:C:\Path\To\dbexport.bacpac /tsn:localhost /tdn:dbname /p:storage=file