He is using LibGDX and pretty much everything in most games developed with LibGDX run on all platforms (iOS, Android, Desktop Java). Most LibGDX games will work in a web browser too (converts to GWT + WebGL). A couple native extensions like some of the freetype stuff won't work in the web environment.
You can write the game on Windows if you want but to compile and test the iOS build you need a Mac with Xcode.
Almost all your game development is done is a project that is shared by all the platforms. During development you typically test the Java desktop version so you don't have to deal with deploying to devices or slow emulators.
Honestly the biggest pain is when you start doing ads or game center type stuff that is platform specific but there are ways to work around that.
You can write the game on Windows if you want but to compile and test the iOS build you need a Mac with Xcode.
Almost all your game development is done is a project that is shared by all the platforms. During development you typically test the Java desktop version so you don't have to deal with deploying to devices or slow emulators.
Honestly the biggest pain is when you start doing ads or game center type stuff that is platform specific but there are ways to work around that.