When using NuGet packages from an external authenticated feed during a build on TeamCity, the credentials for connecting to that feed have to be specified.
Adding this information to source control is not a secure practice, so TeamCity provides the NuGet Feed Credentials build feature which allows interacting with feeds that require authentication.
When editing the build configuration, from the list of available Build Features select NuGet Feed Credentials. In the dialog that is opened, specify the feed URL and the credentials to connect to the feed.
You can add this build feature to any build configuration, one for every feed that requires authentication.