How do I get link URLs from the WordPress links backend into an array?

I believe that you’re looking for the get_bookmarks() function, which returns an array of bookmark objects. You could then implement this into your code:

<?php // Get RSS Feed(s)
include_once(ABSPATH . WPINC . '/feed.php');

$bookmarks = get_bookmarks();
$rsslist = array();

foreach ( $bookmarks as $bm ) {
    if ( $bm->link_rss )
        $rsslist[] = $bm->link_rss;
}

$rss = fetch_feed( $rsslist );

if ( ! is_wp_error( $rss ) ) {
    $maxitems = $rss->get_item_quantity(25); 
    $rss_items = $rss->get_items( 0, $maxitems ); 
}