Save or update data when custom post published

How to save them when any of them are checked

You can use update_post_meta(), like so:

add_action( 'save_post', 'op_save_offers_meta', 10, 2 );
function op_save_offers_meta( $post_id, $post ) {
    // Make sure that we can save the meta data.
    if ( ( empty( $post_id ) || empty( $post ) || ( 'listing' !== get_post_type( $post ) ) )
    || ( empty( $_POST['post_ID'] ) || $_POST['post_ID'] != $post_id || ! isset( $_POST['offers_meta'] ) )
    || ( empty( $_POST['offers_meta_nonce'] ) || ! wp_verify_nonce( $_POST['offers_meta_nonce'], 'save-offers-meta' ) )
    || ( ! current_user_can( 'edit_post', $post_id ) ) ) {
        return;
    }

    $offers_ids = wp_parse_id_list( $_POST['offers_meta'] );
    if ( ! empty( $offers_ids ) ) {
        update_post_meta( $post_id, '_offers_ids', $offers_ids );
    } else {
        delete_post_meta( $post_id, '_offers_ids' );
    }
}

I have also altered your op_render_menu_meta_box():

function op_render_menu_meta_box( $post ) { // Metabox Callback Function Called
    // Get the selected 'offers' (array of post IDs).
    $offers_ids = (array) get_post_meta( $post->ID, '_offers_ids', true );

    // Displays a nonce field, without referrer field.
    wp_nonce_field( 'save-offers-meta', 'offers_meta_nonce', false );

    // Metabox content
    $getPostsToSelect = get_posts(
                        array(
                            'post_type' => 'offers',
                            'posts_per_page' => 3,
                        ));
    foreach ($getPostsToSelect as $aPostsToSelect) {
    ?>
    <label>
        <input 
          type="checkbox" 
          name="offers_meta[]" 
          class="postsToSelect"
          value="<?php echo $aPostsToSelect->ID ?>"
          <?php checked( true, in_array( $aPostsToSelect->ID, $offers_ids ) ); ?>
         /> 
        <?php echo $aPostsToSelect->post_title;

?>
    </label><br/>
    <?php
}
}

and then use or echo those checked offers

You can use get_post_meta() to retrieve the ‘offers’ (i.e. array of post IDs) — just like in the above op_render_menu_meta_box() function, and do a standard The Loop to display the post details. Example:

$post_id = get_the_ID();

// Get the selected 'offers' (array of post IDs).
$offers_ids = (array) get_post_meta( $post_id, '_offers_ids', true );

$q = new WP_Query( [
    'post_type' => 'offers',
    'post__in'  => $offers_ids,
] );

if ( $q->have_posts() ) {
    while ( $q->have_posts() ) {
        $q->the_post();
    ?>
        <h3><?php the_title(); ?></h3>
        <p><?php the_content(); ?></p>
    <?php
    }
}

wp_reset_query();