The number of good reasons why you should analyze your log files would fill many forum topics :) You can usually find anything you're looking for in the log files, but it does come back to your question, what are you looking for?
The log file is normally the only pure feedback you get from visitors. It tells you things about all of the visitors your site gets, even those that just look at one page and leave. Knowing your customers is fundamental to any organization. By using common log analyzers you can find out things like what time of day your site is busiest, how much traffic you're putting through, what day/week/month you're busiest. It will help you predict when you might need to make changes to your server/hosting for bandwidth reasons.
I know I found out that my 2nd most popular page was a webcam that had no linking structure around it, so many visitors would follow links into it and then have no way to navigate the rest of my site. You can find out what the most common exit page is (last page people view before leaving). Though those stats won't necessarily be 100% accurate it can give trends and let you see if there might be a problem with the "flow" of your website.
Finding out how many pages per visitor can be important as well as which pages they view. Are there any pages you put a lot of work into that nobody views? It's helpful to know things like this so you can try to find out why.
In the end, you have to do a lot of thinking and find out what's important to you to help make your site more successful. Log analyzers will help take raw data and make it more readable, but no program will show you the "silver bullet" answers to help make your site a better place to visit. You'll have to work on that one yourself :)
I'm relatively new to the webmaster game, but I come from a sysadmin background so I had a pretty good understanding of exactly how important log files were, but I'm still amazed at what I can learn about my traffic every day when I comb through the logs.
I have my stats generated pretty much on the fly (add any new traffic every 5 minutes) because I like to be able to see roughly real time stats when I'm looking for something. Tha being said, I really only look at my stats long and hard about once to twice a day.
If this doesn't answer enough, ask more :) We're all happy to help.