Added some comments to the function so its easier to follow whats going on.

functions.php

<?php //Do not add the opening PHP tag if you already have one

function my_print_tickets_left_after_selector($atts) {
global $wpdb;
//Set $EVT_ID to the id set within the shortcode
$EVT_ID = $atts['evt_id'];

 $x = '';
    //Setup $x to start UL.
    $x .= '<ul style="margin-left:0;">';
        
        //SQL Query to pull in DTT_name, DTT_reg_limit and DTT_sold values for the event id give.
        $sql = "SELECT DTT_name, DTT_reg_limit, DTT_sold ";
        $sql .= "FROM {$wpdb->prefix}esp_datetime ";
        $sql .= "WHERE {$wpdb->prefix}esp_datetime.EVT_ID = %d";
        
        //Run the query to pull the above details.
        $datetimes = $wpdb->get_results( $wpdb->prepare( $sql, $EVT_ID ));
        
        //Loop over $datetimes, for each element use $datetime within the loop.
        foreach($datetimes as $datetime){
            //If DTT_Name is set, use that value, if not use 'this event'.
            $name = !empty($datetime->DTT_name) ? ($datetime->DTT_name) : 'this event';
            
            //Set $limit to the DTT_reg_limit value.
            $limit = $datetime->DTT_reg_limit;
            
            //Set $sold to the DTT_sold value.
            $sold = $datetime->DTT_sold;

            //Remainder places = $limit - $sold
            //So this is remaining spaces on the Datetime.
            $remainderp = $limit - $sold;

            //Set $remain to with 'bucketloads of' or the value within $remainderp depending if the $limit value it -1.
            //In otherwords, if there is no limit set on the datetime, set $remain to be 'bucketloads of', otherwise $remain should be the remaining spaces.
            $remain = $limit == -1 ? 'bucketloads of' : $remainderp;
            
            //Is $limit equals $sold then the datetime has no remaining spaces, its sold out.
            //So if that's the case we don't want to output anyting to do with remaining spaces, just say its sold out...
            if ( $limit == $sold ) : 
                $x .= $name . ' is <b>sold out</b>';
            //If its NOT sold out, output the detais below:
            else :
                //_nx($single, $plural, $number, $context, $domain)
                //$single is the output when $number == 1.
                //$plural is the output when $number > 1.
                //$number (in this case spaces remaining).
                //$context, used for translation and allows you to set when the translate.
                //$domain, used in translation to say which text domain to use for the translation.
                $x .= sprintf( _nx(
                    '%1$sThere is one space left for %3$s</li>', 
                    '%1$sThere are %2$s spaces left for %3$s</li>', 
                    $remain, 'event ticket info', 'event_espresso' ),
                    '<li style="list-style-type:none;">', $remain, '<span>' . $name . '</span>'
                    );
            endif;
        }
    //Close the UL.
    $x .= '</ul>';

    //Return he output to the page.
    return $x;

}
add_shortcode('show_capacity','my_print_tickets_left_after_selector');