For example, if you have FTP software on your computer to turn that device into an FTP server, you'd install the DDNS application on that computer. That computer is the one that users will reach when they request your server, so it's the one that needs to always be updating the DDNS provider with its current IP address.
What the software does is monitors the dynamic IP address for changes. When the address changes (which it eventually will, by definition), the software contacts the DDNS service to update your account with the new IP address.
This means that so long as the DDNS software is always running and can detect a change in the IP address, the DDNS name you have associated with your account will continue to direct visitors to the host server no matter how many times the IP address changes.
The reason a DDNS service is unnecessary for networks that have static IP addresses is because the domain name doesn't need to know what the IP address is after it's initially told of it the first time. This is because static addresses don't change.
A DDNS service is really only part of the equation when serving files through the internet from your computer. You may also need to tell your router which computer on the network should be contacted when a user outside of the network tries to access the server. This is done by enabling port forwarding on your router.