Great question! There are multiple use cases here.
The primary reason you might want to add a private IP to your Linode is so that it can communicate directly with other Linodes within the same data center that also have private IPs. This can be useful for transfering data securly between your Linodes or even increasing the speed by which the servers communicate with each other. Here's our guide on how to add private IPs that includes some good rules to follow when setting up private networking:
Another reason you might need to add a private IPs to your Linodes is for use with a NodeBalancer. While often used for redundancy and load balancing purposes, NodeBalancers can also even be used for masking public IP addresses. You can read more about NodeBalancers here:
If you need to delete a private IP after-the-fact, the Support Team can help with that. Just open up a ticket requesting the IP to be removed, and we'll take care of it for you. :)