Authorization has been denied for this request - New Web API Project

In the ValuesController there is an attribute Authorize if you remove it, then it will work as home page.

The Authorize attribute just prevent an anonymous users from accessing the ValuesController.

to work with this attribute, you need first to register a user, and then login to get user's token, then you can use the token to authorize your self and get access .

In this page Individual-accounts-in-web-api is explained all what do you need

It happens because you have an Authorize attribute on your ValuesController

[Authorize]
public class ValuesController : ApiController

Just remove [Authorize] and try again

EDIT

According to your edit: You should create a new user and login or use [AllowAnonymous] as mentioned by @Marcus H. Read more about Identity