How do I display offsite database info on my wordpress site?

As @s_ha_dum mentioned in the comments, building an API or some sort of XML or JSON feed (or leveraging one that’s already there, or installing a plugin on whatever system that is, etc) is really your only way of getting at that data remotely. Once you get that going, you can use wp_remote_get to query the API.

To make this question a little more WordPress specific, let’s say you made a JSON feed at domain.com/games.json. In order to get it from within WordPress, you might do something like this:

$response = wp_remote_get( 'http://domain.com/games.json' );
if ( !is_wp_error( $response ) ) {
    $data = json_decode( $response->body ) );
    # Do something with $data
}