For example, say you have a news article that you would like to link to in a blog post, a tweet on twitter, in an IM or Facebook status, in an email, in a newsletter, or in an IRC topic. Most of those have character limits, so the URL would be too long for posting.
For those that don't have character limits, using a short URL can add context to the link, can be communicated verbally, and would look much cleaner.
So, here's an example. The external link you have could look something like this:
And can be changed to something like this with 301tool: