One should note that this is the "new" ASP.NET. The old version, the one explicitly tied to IIS, is not and will probably never be open source software.
They're building a new stack from the ground up. Which is the only way, really, to make it "cross platform".
Not quite: that's "ASP.NET MVC", the razor view engine, "web pages", and some other libraries like Entity Framework and SignalR. Technically these are just libraries built on the closed-source ASP.NET stack, colloquially known as System.Web, which is essentially IIS.
They're building a new stack from the ground up. Which is the only way, really, to make it "cross platform".