checkedListBoxBids.CheckedItems
will return collection of checked items. based on your need you can get each checked item or just one of item like below
foreach(object itemChecked in checkedListBoxBids.CheckedItems)
{
DataRowView item = itemChecked as DataRowView;
string listingID= item["ListingID"];
}
assume you bind as below
checkedListBoxBids.DataSource = dt;
checkedListBoxBids.DisplayMember = "ListingName";
checkedListBoxBids.ValueMember = "ListingID";