The idea mentioned there, is that when a link is followed to a site, for instance, tinyurl.com it gets some value, or "juice", if that site redirects the link with a 301 to another site, it passes some of the value to that other site (final destination of the link).
Is it valuable?, not much.
Using a shortener when you need to save space is very good, but that's it, there is no reason to used them if you can avoid it. There are many drawbacks that also affect the value of your site and links to use a shortener just for the sake of it.
There is a video on Youtube, Are shortened URLs treated differently than other redirects? that has been linked on some resources related to the subject where Matt Cutts talks briefly about that. As he mentions, those links are treated as any other link, which means that they keep some value.
Update, simplified version
Every site has a value, consider each link on site A that points to a site B a little transfer of value, like money on people's account.
When you click a link, you are clicking a fragment of text that describes what is going to be on the other side of the link, so when you click a link on Site A because they mention that there is something interesting on site B, site A is sending a bit of value to Site B.
If the piece of text that you clicked on site A doesn't have the address of site B, but a URL from a shortener service, then Site A passes some value to the shortener service, and then the shortener service, who is doing a 301 redirection to site B, passes some value to them.