They are basically using PHP short tags with a short cut for printing to screen.
<?=get_user_id("SELECT user_id FROM users WHERE auctionid='".$_name['name']."'"); ?>
Is actually just:
<? echo get_user_id("SELECT user_id FROM users WHERE auctionid='".$_name['name']."'"); ?>
Due to possible conflicts with XML, and the fact that the normal PHP tags are always available but short tags may not be, I recommend not using the short tags. Just write:
<?php echo get_user_id("SELECT user_id FROM users WHERE auctionid='".$_name['name']."'"); ?>
As for the shortened calls to the database: you are just looking at a function that does the work. Nothing spectacular at all.