So I understand what a static method or field is, I am just wondering when to use them. That is, when writing code what design lends itself to using static methods and field
I would say use static methods whenever you have functions which are independent of the state of the instance, ie. doesn't depend on any instance fields.
The less non-local state that a method depends on, the easier it is to understand, so static is a helpful signal to the reader of the code.