In order to use Instagram API, you'll have to create Instagram profile and register for developer options on https://www.instagram.com/developer/.
After that, create your API client following Instagram documentation(add few Sandbox users, set Redirect Uri to "https://www.instagram.com", uncheck Disable implicit OAuth etc.)
Copy your Client ID in LoginPage.xaml.cs -> OnInstagramClicked func ->
OAuth2Authenticator auth = new OAuth2Authenticator( clientId: "YOUR_CLIENT_ID", scope: "basic", authorizeUrl: new Uri("https://api.instagram.com/oauth/authorize/"), redirectUrl: new Uri("https://www.instagram.com/"));