What does the "EU Internet" switch do on my iPhone 5s?
I believe it allows data roaming, but only within European Union countries. See http://www.cultofmac.com/288360/europeans-new-roaming-options/:

Inside cellular settings, Apple has added a dedicated “EU Internet” switch alongside the existing data roaming toggle. It gives users the ability to active Internet connectivity when they visit countries inside the European Union, while in other parts of the world in will remain inactive.
Presumably, this change follows new European legislation that says carriers can no longer charge customers for data roaming when they visit other EU countries. The rules don’t officially come into effect until 2015, but a number of carriers have already begun abolishing roaming fees.