Save multiple choice values in SharePoint ListItem

You will find that your checkbox list is populated with the selected value.
Save selected checkboxes in the list item as below

SPFieldMultiChoiceValue multiValue = new SPFieldMultiChoiceValue();
 foreach (ListItem item in multiValueInputFormCheckBoxList.Items)
 {
     if (item.Selected) multiValue.Add(item.Value);
 }
 item["multivalued choice field name"]= multiValue;

Happy Coding !!!