Technorati specializes in indexing the world of blogging, providing search engine services and other facilities meant to make it easier to traverse blogging. Of course a big feature of blogs is the syndication (RSS or Atom) which allows software to easily determine what is new on a site. Technorati of course triggers its indexing based on the RSS feed a site publishes.
Pinging is a common method in blog aggregation sites. The ping is a little message sent to the aggregator that says "I have new content" and the aggregator is expected to collect that new content and aggregate it. With technorati this feature is discussed here, but note that page is very developer centric. It's assumed you are using some kind of publishing software, and that the publishing software handles the pings. And, indeed, most of the publishing software does handle pings for you, and all one needs do is enter a ping URL into their publishing software. For technorati that ping URL is documented, though unclearly, as being:
Claiming your blog is a technorati specific method that isn't as fast as causing your blog software to send a ping to technorati. The ping method is proactive in telling technorati specifically when you have new content, and the expectation technorati gives is that their servers will quickly consult your servers for the new content.
When you claim a blog there are several things that go on, one being that technorati will schedule regular visits to your RSS feeds to check for new content. So if you're unable to configure your software to ping technorati, they'll still find your new content and index your site. The difference is that it will be on their schedule rather than your schedule.