drupal_get_form causes page to be reinvoked?

Once you change your function declaration to

function download_report($form_state) {

If you add as the first line

drupal_set_message('Values:<pre>' . print_r($form_state['values'] . '</pre>');

when you submit the form what shows for the values?

Hi,
Drupal's request dispatcher mechanism is referred to as a "menu". The "menu" accepts/translates a user request like http://my_drupal_site/request_report and then dispatches that request to your application code.
.....
Drupal performs all sorts of infrastructure processing to make sure you form and application logic is secure. All you have to do is request Drupal to perform the infrastructure processing. You are very close. See code example below (call drupal_get_form from your "menu").
.....
Drupal also makes heavy use of naming convention to identify modules and functions. Therefore, it's critical that you use the correct naming pattern for Drupal to identify, and thus call your application code.
.....
The acronym you are looking for (in the the Drupal documentation jargon) is FAPI.
http://api.drupal.org/api/drupal/developer--topics--forms_api_reference....

Naming examples:
.....
Module: my_download_report
Menu: my_download_report_menu
Form: my_download_report_form
Form Validation: my_download_report_validation (not demonstrated here)
Form Submission: my_download_report_submit
......
Use the submission phase to save the user's date selection. It's a hack, since you aren't persisting anything. We'll persist the user's selection in the session object.
......
Your sql is not using the date values?? Why have the user's set date values if you report doesn't use them????
....
If you want to display the report results and form on the same page, you can use the form's #prefix and #suffix attributes. Or, you can add a new display only form element ('#type'=>'markup', '#value'=>$YOUR_HTML_MARKUP).
.....
You are also missing some logic in your database processsing. After you call "db_query()", you need to extract the results into a php variable for storage and/or output:

$out = '<table cellspacing="3" cellpadding="3">';

// NOTE! We have to loop through the results

while($my_result = db_fetch_object($result)) {
     $num_downloads = $my_result->downloads;
     $file_name = $my_result->filename;
     $out .= '<tr><td>'.$num_downloads.'</td><td>'.$file_name.'</td></tr>';
} 
$out .='</table>';

..............................
Here's the whole hack:

Note! You have to use the adminstration pages and clear all cache to get drupal to recognize the new "menu". Once cache is cleared, request your report with the web browser set to http://my_drupal_site/my_download/request_report

function my_download_report_menu() {
    $items['my_download/request_report'] = array(
         'type' => MENU_CALLBACK,
         'access arguments' => array('access content'),
         'page callback' => 'drupal_get_form',
        'page arguments' => array('my_download_song_date_range'),

      );
     return $items;
}


function my_download_report_form_submit($node, &$form) {
   // Drupal sets all submit buttons to name=op 
   
   $operation=  $form['values']['op'];
   switch($operation) {
        case 'submit':
            $_SESSION['my_download_report_start_date']  = $form['values']['start_date'];
            $_SESSION['my_download_report_end_date']  = $form['values']['end_date'];
        break;
        case 'reset':
            unset( $_SESSION['my_download_report_start_date']);
            unset( $_SESSION['my_download_report_end_date']);
        break;
   }
}

function my_download_report_form($node, &$form) {

     if ($_SESSION['my_download_report_start_date']) {
	  $default_start = $_SESSION['my_download_report_start_date'];
    } else {
	  $d = date_array(date_now());
	  $default_start = date_make_date($d['year']."-01-01");
    }

     if ($_SESSION['my_download_report_end_date']) {
	  $default_end = $_SESSION['my_download_report_end_date'];
     } else {
	  $d = date_array(date_now());
	  $e = date_make_date(($d['year']+1)."-01-01");
          $default_end= date_format_date($e, 'custom', 'Y-m-d');
     }
  
     $output = '';

     if ($_SESSION['my_download_report_start_date'] && $_SESSION['my_download_report_end_date'])
     {
           // if the user clicked submit run the report 
           // it appears as thouh the date values above aren't used in the database query?? That seems like another bug.
          $output = _run_report();
  }
 
    $form['report'] = array(
         '#type='=>'markup',
         '#value'=>$output,
         '#weight'=>1,
  };

    $form['start_date'] = array(
       '#type' => 'date_popup',
       '#title' => t('Start Date'),
       '#default_value' => $default_start,
       ' #date_format' => 'Y-m-d',
       '#prefix' => '<table><tbody style="border-top:0;">' . '<tr><td>',
        '#suffix' => '</td>', 
   );

   $form['end_date'] = array(
      '#type' => 'date_popup',
       '#title' => t('End Date'),
       '#default_value' => $default_end,
       '#date_format' => 'Y-m-d',
       '#prefix' => '<td>',
      '#suffix' => '</td>',
    );

    $form['submit'] = array(
        '#type' => 'submit',
       '#value' => t('Generate Report'),
       '#prefix' => '<td>',
       '#suffix' => '</td></tr></table>',
     );
   
   // allow the users to clear the session values 
     $form['reset'] = array(
         '#type' => 'submit',
         '#value' => t('reset'),
       );

      return $form;

}

// Note! Verify the sql works outside of this code. I can't tell without the actual data or schema 

function  _run_report() {

    $result = db_query( "SELECT count(tbl.uid) downloads, filename
      FROM ( SELECT distinct uid, fid FROM downloads ) tbl
      JOIN files ON (tbl.fid = files.fid) GROUP BY tbl.fid"
    );
  
   $out = '<table cellspacing="3" cellpadding="3">';
    while($my_result = db_fetch_object($result)) {
      $num_downloads = $my_result->downloads;
      $file_name = $my_result->filename;
      $out .= '<tr><td>'.$num_downloads.'</td><td>'.$file_name.'</td></tr>';
    } 
    $out .='</table>';
  
    return $out;
}

.
....
Hope that helps.