How to send a notification e-mail to the admin when a registration has been made?

At the moment, there is no mechanism in place to send emails to an administrator when a registration is made.

If you wanted to be able to customize the message sent out in the Drupal admin, you could implement the following hook:

hook_recurring_events_registration_notification_types() to add a new notification type, for example 'admin_registration_notification'. See the recurring_events_registration.module file for how this is done.

/**
 * Implements hook_recurring_events_registration_notification_types_alter().
 */
function recurring_events_registration_recurring_events_registration_notification_types_alter(array &$notification_types) {
  $notification_types += [
    'admin_registration_notification' => [
      'name' => t('Admin Registration Notification'),
      'description' => t('Send an email to an admin when someone registers for an event?'),
    ],
  ];
}

Then you could probably implement hook_ENTITY_TYPE_insert() with ENTITY_TYPE being 'registrant', with something like this:

use Drupal\Core\Entity\EntityInterface;
 
/**
 * Implements hook_ENTITY_TYPE_insert().
 */
function YOUR_MODULE_registration_insert(EntityInterface $entity) {
  $key = admin_registration_notification';
  $to = 'your-admin-email@domain.com';
  $mail = \Drupal::service('plugin.manager.mail');
  $params = [
    'registrant' => $entity,
  ];
  $mail->mail('recurring_events_registration', $key, $to, \Drupal::languageManager()->getDefaultLanguage()->getId(), $params);
}

Thanks for this fast and great feedback!

I had to correct one thing

function YOUR_MODULE_registration_insert(EntityInterface $entity) {

It does not have to be registration_insert, but registrant_insert, then it works.

use Drupal\Core\Entity\EntityInterface;
 
/**
 * Implements hook_ENTITY_TYPE_insert().
 */
function YOUR_MODULE_registrant_insert(EntityInterface $entity) {
  $key = 'admin_registration_notification';
  $to = 'your-admin-email@domain.com';
  $params = [
    'registrant' => $entity,
  ];

  $mail = \Drupal::service('plugin.manager.mail');
  $mail->mail('recurring_events_registration', $key, $to, \Drupal::languageManager()->getDefaultLanguage()->getId(), $params);
}

Ah yes, sorry. Great catch. I am glad it works for you!