How do I open SSRS (.rptproj) files in Visual Studio 2013?

It can be opened with Microsoft SQL Server Data Tools - Business Intelligence for VS2013:

http://www.microsoft.com/en-us/download/details.aspx?id=42313

The download is live again as of 6/2/14

SSRS is not a part of VS2013, it is part of SQL Server. You need the BI Dev Studio installed in order to be able to open SSRS projects (rptproj). It uses VS2013 just as a shell similar to what other products do...

You should install Business Intelligence Studio, it comes as part of MS SQL Server installation.

Business Intelligence Development Studio (BIDS) — is a part of MS SQL Server. BIDS is a IDE for rdl reports development based on Visual Studio Shell. BIDS allows you to open .rptproj files. If you install BIDS on the same box with your regular Visual Studio, you will be able to open .rptproj and .csproj files from one IDE.

The issue is that SQL Server installations include BIDS based on previous version of Visual Studio Shell, i.e. SQL Server 2008 R2 will provide you BIDS integrated to Visual Studio 2008, while SQL Server 2012 BIDS will be based on Visual Studio 2010.

As far as I know, there is no BIDS that integrates into VS2013.

There is version for VS2012.

SQL Server 2014 will ship BIDS based on VS2012 as well.

I believe we will see BIDS for VS2013 not earlier than in SQL Server 2016.