Alter form after submit
hook_form_alter is probably a good approach for displaying the search words in the form.

Assuming you're not submitting the form via AJAX, you need a way of passing the search words to the next page load. One way might be to redirect to the same URL but adding your search words as URL query parameters. You could do that by setting $form_state['redirect'] or calling drupal_goto in your submit function.

Another way, maybe easier, would be to save your search words in the user's session and reading them back in the subsequent page.