No-Ip Question

I've looked all over Google and Null-Byte about figuring out what No-ip is really doing to my ip on the internet and I can't gain clarity.

From my understanding, no-ip is just a dynamic DNS service that takes my ip (whatever it is) and links it to a name (fox.no-ip.com). However, I just don't understand how this can help me mask anything. I can go to ipchicken.com and I just see my ip. Nothing changes.

I've read on How-To's here and it mentions running no-ip AFTER going through a VPN and I just don't understand the benefit in doing it. Is it to prevent some sort of leak? Any information is appreciated.