"When I requested the Key I provided application site. What is purpose of this site ? Can it be just my home page, git account or a personal blog ? Or do I need to implement some functionality on this site ?"
As far as I know, the application site field exists purely for documentational purposes. You can provide the name of your blog or a GitHub page if you wish. You'll want to link to something that users of your application can visit for more information about the app.
"I got a Client Id, Client Secret and Key. If I understand correctly I need to use only key ? In which cases do I need the Client Secret ?"
The API key is used to make requests for all of the public-facing information on the site (with the exception of the /events route). Unless you are implementing authentication in your application, you will only ever need the API key.
The client ID and secret are used for authenticating users and you can find more information about their usage here.
"Do I need to do something after I get the Key to make it works ? I am using Rubyoverflow library, but I got error message when I set my key ("Invalid key")"
No, your key should work immediately. Are you sure you have entered the key exactly the way it is listed on the application's settings page? You might want to contact the author of the RubyOverflow library to find out exactly what causes such an error.
