I think you need something like this:
var checkedValues = strings.Where(s => IsChecked(s))
.Select(s => "'"+s+"%'")
.ToArray();
bindingSource.Filter = "LoadName LIKE " + string.Join(" or ",checkedValues);
Where IsChecked()is the fake method, you should replace it with something telling you if the value is marked or not in yours CheckedListBox.
, CheckedListBox - , :
strings.Where(s => IsChecked(s))
:
ceckedListBox.CheckedItems.Cast<string>()