I won’t make any excuses; I love vi. Folks at The Register called vi Bill Joy’s greatest gift to mankind (who am I to argue?) However, being able to use vi well goes beyond whether you like it or not.
Why should you know vi (well)? For one thing, it is on every UNIX and Linux system. Emacs certainly isn’t, and neither are the other worthy competitors like pico, elm, nano, and others. None of these editors, good though they may be, are ubiquitous enough. In fact, every last one requires special installation on virtually every platform – that is, there are no default installations that include them.
Even vim is not ubiquitous – although it may be available for lots of environments, it is not necessarily installed by default. For a long time, I used vim in vi-compatible mode for just this reason. (My favorite vim-specific idea? :set nowrap).
Vi has an interesting history. Vi was written by Bill Joy, starting with the basics of the em editor from George Coulouris of Queen Mary College, University of London. Bill wrote vi to be able to work at 300 baud (as he says, that’s all he had and the fellows at MIT doing EMACS had the equivalent of fiber for that day and age). UNIX Review did an interview with Bill in August of 1984.
The original vi is now available under a BSD license. In contrast to EMACS and vim, both of which number in the multi-megabytes, vi was 160k on the i386.
So what editor does Bill use? ed.
UPDATE: Eric Raymond in his book The Art of UNIX Programming (also available online) has a very interesting analysis in Chapter 11 of the complexity and contrast (and psychology) between EMACS and vi.
UNIX Administratosphere 
Is this a good reason to know vi though?
Definitely. As an administrator, you will be faced with boxes that have the minimum installed, or are a different operating system than you are used to. Many systems also are restricted, and may not allow the addition of third party editors.
However, vi will always be there – so you should know vi even if you don’t use it every day.