You need the 2 digit month number that you want to query on and then use the code below. This should be easy with php (for example, see this post). In the code below $month
is the number of the month in a 2 digit format, eg March would be 03.
$start_date = date('Y'.$month.'01'); // First day of the month
$end_date = date('Y'.$month.'t'); // 't' gets the last day of the month
$meta_query = array(
'key' => 'event_start_date',
'value' => array($start_date, $end_date),
'compare' => 'BETWEEN',
'type' => 'DATE'
);