For Android, Xamarin runs the Mono Runtime side-by-side to the Dalvik (and/or Android) Runtime.
I recommend checking out this doc:
For iOS, Xamarin uses an AOT (Ahead of Time) compiler.
Unfortunately, I don't have a good link to send you that goes into more detail.
And if you're interested in metrics check out this blog post by an ex-googler:
Those metrics are quite flawed though, J2Obj-C(Java auto-translated to Obj-C) outperforming Obj-C is a pretty clear sign that they are comparing apples and oranges, Xamarin however does perform reasonably well on both iOS and Android as long as you stick with the native API wrappers and don't use the horribly buggy and slow Xamarin.Forms(allthough being able to share your UI code between WP,Android and iOS might make it worth the hassle to get that junk working decently).