The hidden search object is an internal singleton instance that is used by all the result web parts on the search result page. The web parts pass properties and settings to the hidden object and use it to handle all query specific logic.
Since the class is marked as internal it is only accessible within the search assembly. It cannot be used and specialized in custom web parts.
The hidden object parses the HTTP Post/Get information of the request and runs the query using the SharePoint object model - either using the FullTextSqlQuery or the KeywordQuery class.