What does the "EnableViewState" property do in ASP.Net Controls? Why would I want it on or off?

It enables the ViewState on the page. It allows the page to save the users input on a form.
Enable ViewState Property turns on the automatic state management feature that enables server controls to re-populate their values on a round trip without requiring you to write any code. This feature is not free however, since the state of a control is passed to and from the server in a hidden form field. You should be aware of when ViewState is helping you and when it is not.
For example, if you are binding a control to data on every round trip, then you do not need the control to maintain it’s view state, since you will wipe out any re-populated data in any case. ViewState is enabled for all server controls by default. To disable it, set the EnableViewState property of the control to false.