How does a rep take ownership of the call from an irate customer?

The key is to first calm the customer down, until you do this they are less likely to listen to you. Also remember that, in most cases, they will be angry with the customer, not you. With this in mind you can disassociate yourself from the company slightly, enough to remind the customer you are a person, not the company but you will still be able to help them with their problem.

One method is to sympathise with the customer first using lines such as "I can understand how inconvenient [state their problem] must have been for you". It also helps to ensure they know your name and if you have to investigate and call them back that they are reassured that you are actually going to call them back.