iOS app development in visual studio 2013

You can sort of do this with Xamarin, as well as building Android applications. You still need to have a Mac in order to build iOS/Mac OS X applications, but you don't need to interact with it directly - it just needs to be on the network in order to complete the build the app, for licensing reasons. On the other hand, to run/debug the app, you'll either need a device plugged into your Mac, or run the emulator on your Mac. (Personally I have a Mac Mini for this purpose, and it's on my KVM... so I don't use it directly very often, but it's available when I need it.)

This is not building code in Swift or Objective-C - it's using C# which is then ahead-of-time compiled into native code. The Cocoa Touch libraries are exposed to your C# code so you're still using the same widgets etc... you're just building in C#.