I had the same problem and I found the answer in the PHPMailer documentation.
PHP 5.6 certificate verification failure
In a change from earlier versions, PHP 5.6 verifies certificates on SSL connections. If the SSL config of the server you are connecting to is not correct, you will get an error like this:
Warning: stream_socket_enable_crypto(): SSL operation failed with code 1. OpenSSL Error messages: error:14090086:SSL routines:SSL3_GET_SERVER_CERTIFICATE:certificate verify failed
The correct fix for this is to replace the invalid, misconfigured or self-signed certificate with a good one. Failing that, you can allow insecure connections via the SMTPOptions property introduced in PHPMailer 5.2.10 (it's possible to do this by subclassing the SMTP class in earlier versions), though this is not recommended:
$mail->SMTPOptions = array( 'ssl' => array( 'verify_peer' => false, 'verify_peer_name' => false, 'allow_self_signed' => true ) );
You can also change these settings globally in your php.ini, but that's a really bad idea; PHP 5.6 made this change for very good reasons.
Sometimes this behaviour is not quite so apparent; sometimes encryption failures may appear as the client issuing a QUIT immediately after trying to do a STARTTLS. If you see that happen, you should check the state of your certificates or verification settings.