How do I use $form_state to return data from the from submission handler to the form builder?
The only functions that get $form_state are form builders, form validation handlers, and form submission handlers. 

To say the truth, your code doesn't need to use MYMODULE_form_page(): You can simply use the form builder function.

function MYMODULE_menu() {
  // Module settings.
  $items['MYMODULE'] = array(
    'title' =&gt; 'MYMODULE',
    'description' =&gt; 'MYMODULE',
    'page callback' =&gt; 'drupal_get_form',
    'page arguments' =&gt; array('MYMODULE_form'),
    'access callback' =&gt; TRUE,
  );
  return $items;
}

function MYMODULE_form($form, &amp;$form_state) {
  $form['header_text' =&gt; array(
    '#markup' =&gt; '&lt;p&gt;&lt;/p&gt;',
  );

  $form['kname'] = array(
    '#title' =&gt; t('What would you like to search'),
    '#type' =&gt; 'textfield',
    '#required' =&gt; TRUE,
    '#description' =&gt; t('Please insert the keyword you whish to search for.'),
  );

  if (isset($form_state['MYMODULE_result'])) {
    $form['footer_text'] = array(
      '#markup' =&gt; '&lt;p&gt;'. $form_state['MYMODULE_result'] .'&lt;/p&gt;',
    );
  }

  $form['submit'] = array(
    '#type' =&gt; 'submit',
    '#value' =&gt; 'Send',
  );

  return $form;
}

function MYMODULE_form_submit($form, &amp;$form_state) {
  $keyword = $form_state['values']['kname'];
  $op = MYMODULE_simple_page($keyword);

  // Form submission handlers don't return any value.
  // If you need to return a value to the form builder, you should use the following code.
  $form_state['MYMODULE_result'] = $op;
  $form_state['rebuild'] = TRUE;
}