Components
To access the meta tags field data created by the Metatag module you can use the Metatag Manager service.
$metatag_manager = \Drupal::service('metatag.manager');
First, get the tags from the entity (usually from a node):
$tags = $metatag_manager->tagsFromEntityWithDefaults($node);
and then generate the meta tag values:
$metatags = $metatag_manager->generateTokenValues($tags, $node);
In summary, the Metatag Manager service in Drupal allows for programmatic retrieval of meta tags associated with an entity.