Register Taxonomy – What is `query_var`?

It is explained pretty thoroughly in the Codex, in my opinion. It serves a purpose e.g. when you want to grab posts for your taxonomy, it is a URL slug for your taxonomy, e.g. example.com/tag/cats -> example.com/query_var/cats.

Note: The query_var is used for direct queries through WP_Query like

new WP_Query( array( 'people' => $person_name ) )

and URL queries like /?people=$person_name. Setting query_var to false will disable these methods, but you can still fetch posts with an explicit WP_Query taxonomy query like

WP_Query(array('taxonomy'=>'people', 'term'=>$person_name)).

Basically, query_var is part of the WordPress Query Vars.