How to use Open ID

OpenID is an open web protocol for "federated log-in". From a users practical point of view, this means that

1) you don't need to come up with a new password for every website you register on. Instead you only need a single password for your "OpenID identity provider"

2) With OpenID, OpenStreetMap (and other websites using OpenID) never sees or knows your password. So if OSM ever gets hacked or compromised, if you are using OpenID your password can't be stolen as OSM doesn't have it.

3) Related to 1 and 2) if your password does somehow get compromised, you only need to change it in one place and not on every website you might have used it.

The most commonly used OpenID provider is probably google. I.e. if you have a google account, you can login using that account. Furthermore, if you are already logged into your google account, you can login to OSM with a single button click. No need to type in any password.

However, OpenID being an Open and federated protocol, if you don't trust e.g. google due to privacy concerns, OpenID allows you to choose from a large number of identity providers or even setup your own. So you are not locked in.

Overall, OpenID is really just meant as a simple and convenient way to dealing with the "far too many username and password" problem. It is however completely optional and if you don't want to use it there is no need for it.

As mentioned above, you can find additional information on Wikipedia

OpenID is a protocol for a "federated identity", i.e. you have an account with some identity provider and use that to log in anywhere. Read more on Wikipedia.

the osm.org login page has (well, only if you select the first icon) a nice "what is this" link which links to our wiki page https://wiki.openstreetmap.org/wiki/OpenID

If you mean the login on this help site: it has an info link on the left, two on the right and a short info text on the right.

Is there anything not clear or did you just not see the links?