There are some tips on how to secure your API keys at Using API Keys documentation.
It is stated that embedding API keys directly in the code should be avoided, which is the way you are having it right now, therefore you are getting the warning message.
Follow the tips on that page and you should properly secure your API key. As soon as you do so, the warning will go away.
UPDATE
To avoid having a long discussion in comments, allow me to elaborate further providing this update.
Google provides different ways of authentications to give you more options for securing your apps based on your needs. The warnings are helpful tips to make your apps more secure when going in production or exposing to public. So in your case, it would be better to use a different way of authentication.
API keys can be used in server side. e.g. If you are using an API key to authenticate a 3rd party service from an App Engine app, you can use this key as you already have it, since it is impossible for the key to get exposed. (Avoid using API keys in JavaScript since inspecting the page in the browser will expose the API key as well)
In your case, since you are developing an Android app and/or an iOS app, having the API key in the code is dangerous. Because, anyone can use the .apk or the .ipa file and find a way to access it. Therefore, for developing Android apps and iOS apps it is suggested going with different authentication method. The other authentication method supported in YouTube player API is using OAuth 2.0. For more information you can check the Registering your application documentation.