WP_POSTMETA – What do these values mean inside the data structure?

WordPress will serialize arrays when saving to post meta. The data stored under the key _employer_socials in your example looks like this:

Array
(
    [0] => Array
        (
                [network] => facebook
                [url] => #
        )

    [1] => Array
        (
                [network] => twitter
                [url] => #
        )

    [2] => Array
        (
                [network] => linkedin
                [url] => #
        )

    [3] => Array
        (
                [network] => instagram
                [url] => #
        )
)

When you get the meta, WP will unserialize it for you, and it will look like above:

$employer_socials = get_post_meta( get_the_ID(), '_employer_socials', true );

You can then just work with the data as a regular array. When saving the array, WP will serialize it for you. Here are some basic functions that you can use as a starting point:

/**
 * Updates an existing network, or adds a new one if it does not exist.
 *
 * @param int    $post_id The Post ID to be updated.
 * @param string $network The name of the network to check.
 *
 * @return bool True if an update was made, false otherwise.
 */
function wpse_update_employee_social( $post_id, $network, $url ) {

    $employer_socials = get_post_meta( $post_id, '_employer_socials', true );

    // Update.
    if ( wpse_has_employer_social( $post_id, $network ) ) {
        foreach ( $employer_socials as $key => $employer_social ) {
            if ( $employer_social['network'] === $network ) {
                $employer_socials[ $key ]['url'] = $url;
                update_post_meta( $post_id, '_employer_socials', $employer_socials );
                return true;
            }
        }
    } else {
        // Add.
        $employer_socials[] = [
            'network' => $network,
            'url'     => $url,
        ];
        update_post_meta( $post_id, '_employer_socials', $employer_socials );
        return true;
    }

    // Nothing was updated.
    return false;
}

/**
 * Checks if a given network exists.
 *
 * @param int    $post_id The Post ID to be updated.
 * @param string $network The name of the network to check.
 *
 * @return bool True if the network exits, false otherwise.
 */
function wpse_has_employer_social( $post_id, $network ) {
    $employer_socials = get_post_meta( $post_id, '_employer_socials', true );
    foreach ( $employer_socials as $employer_social ) {
        if ( $employer_social['network'] === $network ) {
            return true;
        }
    }

    return false;
}

Example of adding a new employee social:

wpse_update_employee_social( 5302, 'wpse', 'https://wordpress.stackexchange.com/users/208214/user1669296' );

Example of updating an employee social:

wpse_update_employee_social( 5302, 'facebook', '#1234' );