how to get month and year value
The value for #default_value must have the date() format "Y-m-d H:i:s"; the value returned in $form_state is a string.

In your case, the code you should use is the following one.

  $form['billing']['expiry_db'] = array(
    '#type' =&gt; 'date_select',
    '#title' =&gt; t('Expiry Date'),
    '#default_value' =&gt; date('Y-m-d H:i:s'),
    '#date_format' =&gt; 'm Y',
    '#date_label_position' =&gt; 'within',
    '#date_year_range' =&gt; '0:+20',
    '#required' =&gt; TRUE,
    '#prefix' =&gt; '&lt;div class="field-wrapper-w1"&gt;',
  );


If the form has #tree set to TRUE, then $form_state['values']['billing']['expiry_db'] a string such as '2020-05' for May, 2020.  

As far as I can see, the value of #date_format is only used to understand which components of a date must be returned, but the format of the returned value is always the same. Changing #date_format to 'Y-H:s' the returned value is '2020 00:00'. In you case, the code to get the month, and the year is the following one.

list($year, $month) = explode('-', $form_state['values']['billing']['expiry_db']);


References


_date_api_element_info()