That is the actual Form. To be able to pick and choose which fields  are
on the form  you need to edit  that form in SPD and replace the ListView
Web part  with a DataForm web  part. Simply open a copy of the form for
editing, delete the existing web part, and then click on Insert >
SharePoint Controls > Custom list  Form. That will insert an Appropriate
Data Form web part where you can delete individual fields from the form.
Then save the new form and associate it with the list using the
Supporting files tab on the properties panel for the list.