Linq hides the loops from you, it uses them anyway:
List<String> allFileNames = dtOutput.AsEnumerable()
.Select(r => r.Field<string>("FileName"))
.ToList();
But why do you want a list at all when you already have the table which is also an in-memory collection?
If you want to diplays these file-names in another multiline TextBox
you can use:
txtBoxWriteTo.Lines = dtOutput.AsEnumerable()
.Select(r => r.Field<string>("FileName"))
.ToArray();