How to show post list based on taxonomy term?

In your custom templates, First you have to fetch all terms for your custom post type.

You can use get_terms to retrive terms in taxonomy

If you are using wordpress prior to 4.5

$terms = get_terms( 'competition' , array(
    'hide_empty' => false
) );

OR if you are using wordpress prior to 4.5.0 or later

$terms = get_terms( array(
    'taxonomy' => 'competition',
    'hide_empty' => false,
) ); 

Next Build Term name lists:

$termNames = array();
$count = count( $terms );
if ( $count > 0 ) {
    foreach ( $terms as $term ) {
        $termNames[] = $term->name;
    }
}

Next loop trough each terms and query posts for the current terms & then run the loop:

foreach($termNames as $termName) :

$args = array(
    'post_type' => 'football_fixture',
    'competition' => $termName,
    'order' => 'ASC',
);
query_posts( $args );
if(have_posts()): ?>
    <div class="<?php echo $col; ?>">
        <h2><?php echo $termName; ?></h2>
        <?php while(have_posts()):  the_post(); ?>
        <div id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
             <?php
                $id         = get_the_ID();
                $date       = rwmb_meta( 'pb_match_date','', $post->ID);
                $time       = rwmb_meta( 'pb_match_time','', $post->ID );
                $competition = rwmb_meta( 'pb_match_competition_cats','', $post->ID );
                $team_a     = get_post_meta( $post->ID, 'match_details_home_team', true );
                $team_b     = get_post_meta( $post->ID, 'match_details_away_team', true );

            ?>
            <div class="fixture-item">
                <div class="row">
                    <div class="col-xs-4">
                        <div class="media">
                            <div class="media-body">
                                <h4><?php echo $team_a; ?></h4>
                            </div>
                        </div>
                    </div>

                    <div class="col-xs-4 match-time">
                        <i class="fa fa-clock-o"></i> <?php echo $time; ?>
                    </div>

                    <div class="col-xs-4">
                        <div class="media">
                            <div class="media-body">
                                <h4 class="pull-right"><?php echo $team_b; ?></h4>
                            </div>
                        </div>
                    </div>
                </div>
            </div>
        </div><!--/#post-->
    <?php endwhile; ?>

    <?php 
        wp_reset_query();
    ?>