How to display checkbox meta array values one by one?

Use PHP explode function and convert string into array using a delimiter | then you can use foreach loop.

Example:-

$str = get_the_author_meta('my_pet', $author_id);
$pets_array = explode('| ', $str);

if (is_array($pets_array)) {
    foreach ($pets_array as $pet) {
        echo $pet;
        echo 'somtext';
        echo '<br />';
    }
}