OK had a go at this myself, I didn’t realise ACF was able to add fields to taxonomies which is really handy so I wanted to figure it out too.
<?php
$libargs=array(
'hide_empty' => 0,
'parent' => 0,
'taxonomy' => 'library_categories');
$libcats=get_categories($libargs);
foreach($libcats as $lc){
$termlink = get_term_link( $lc->slug, 'library_categories' );
?>
<a class="single-library-cat" href="https://wordpress.stackexchange.com/questions/71287/<?php echo $termlink; ?>">
<img src="<?php the_field('taxonomy_image', 'library_categories_'.$lc->term_id); ?>" />
<?php echo $lc->name; ?>
</a>
<?php } ?>
It’s in the docs here http://www.advancedcustomfields.com/docs/tutorials/retrieving-values-from-other-pages-taxonomy-user-media/
<?php the_field('taxonomy_image', 'library_categories_3'); ?>
So just replace the field name with your field name and library_categories_ with the taxonomy name. That should do it!