How and why would I "claim ownership" of a paper on arXiv? My colleague has just published a paper on which I am co-author. This is the first paper that has my name on so I am not familiar with the systems yet. 

He has received an email from arXiv to confirm his submission and it gave a "paper password". It recommended he forward this to the co-authors so "They may use it to claim ownership."

What does this mean, why would I want to do it and how do I go about it?
By claiming ownership the paper appears in your arXiv account, and you can manage it there (e.g. uploading a new version or adding a journal reference). It also means it will appear in the list of papers associated to your arXiv author identifier, which is especially useful if you have a common name.

You can claim ownership by logging in to your arXiv account (create one first if you don't have one yet), and then click on the "Claim ownerhsip" link. It will ask for the paper ID and the password.

Edit: As Wrzlprmft pointed out in the comments, it will likely also add to your endorsement record, meaning that you are less likely to need to be endorsed for future submissions you make yourself, and will sooner be able to endorse others.