Try this (using System.Linq):
OfType() is an extension method, so you need to use System.Linq
List<string> selectedFields = new List<string>();
selectedFields.AddRange(chkDFMFieldList.CheckedItems.OfType<string>());
Or just do it in one line:
List<string> selectedFields = chkDFMFieldList.CheckedItems.OfType<string>().ToList();