How would I use this filter to change the output of the date format to “Twitter time”?

WordPress has a core function human_time_diff that does what you want, using it with the filter you provide you have someting like so:

add_filter('latest_tweets_render_date', function( $created_at ){
    $date = DateTime::createFromFormat('D M d H:i:s O Y', $created_at ); 
    return sprintf( '%s ' . __( 'ago' ), human_time_diff( $date->format('U') ) );
});