There is no way for you to verify that, and of course this is what a malicious VPN provider would claim as well. The VPN provider can see all your traffic, and do with it what they want - including logging.
It only applies to using it as a glorified proxy, which is what every third-party 'VPN provider' does. Note: The content in this post does not apply to using VPN for their intended purpose that is, as a virtual private (internal) network. You're probably reading this because you've asked what VPN service to use, and this is the answer.
