when you want to use related lookup field (Interested__c) on any object (contact) then you have to use "Interested__c" field only it gives you only Id of related record (Interested__c).
When you want more information from lookup field like Name and other fields then we have to use lookup relationship object which is "Interested__r".
This object will contains all information related to your lookup record.
It will provide name for related lookup record :
select Id, Name, Interested__r.Name from Contact
Relationship type Name "Contacts" will be used only when you have to get all Interest_Level__c record related to contact.