Remove the Argument from a views URL
The exposed filter form is always going to add a query string to the current URL as it's a simple GET form.

You need a form with a submit handler that redirects to the argument based path. You could do this to the existing exposed filter with hook_form_FORM_ID_alter or create a custom form and expose it in a block.

To make the existing form redirect you'd need to do something like:

function mytheme_form_views_exposed_form_alter(&amp;$form, &amp;$form_state) {
  $form['#method'] = 'POST';
  $form['#submit'] = array('mytheme_redirect_taxonomy');
}

function mytheme_redirect_taxonomy($form, &amp;$form_state) {
  $form_state['redirect'] = '/browse/' . $form_state['values']['category'];
}