Silverlight is a Microsoft product. Hotmail is a Microsoft service. Hotmail probably needs Silverlight for something.
Microsoft Silverlight is a free web-browser plug-in that enables interactive media experiences, rich business applications and immersive mobile apps.
Microsoft Silverlight is a powerful tool for creating and delivering rich Internet applications and media experiences on the Web. Silverlight 5 builds on the foundation of Silverlight 4 for building business applications and premium media experiences.
Edit : I'm assuming you haven't got Silverlight, or that it's disabled on your system. It's safe to download if you haven't got it already, but go to the Microsoft site to get it rather than clicking on some pop-up window (that's just generic stay-safe advice). If it's on your system but disabled in your browser then it's safe to enable it.
Thanks so much! That was the clue I needed. I uninstalled and reinstalled and that cleared it all up.