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Link analysis is the process by which you can see the status of all the links on your site and where they lead to. This can be a useful seo tool for a variety of reasons, but mostly they boil down to one thing. Search engine crawlers hate dead links. They stop the robots from crawling your site efficiently, and they are a marker of a site that either isn't maintained well, or has a sloppy design in the first place. For good search engine optimization, you need to ensure that your site doesn't have any dead links and do good link analysis to make absolutely certain of this.
The other reason why you need to check your links is that with time outgoing links can lead to dead pages. Even worse than this is if a domain that you linked to gets bought without your knowing and turns into a site that you wouldn't want to link to. There's no reason in passing your site rank onto another page if it doesn't contain the same material that it did when you created the link. This may even lead to your being penalized by search engines if the site ends up being not very high quality after it has been taken over.
The bottom line, however, is that your link network needs to be live and lead only to the pages that you want it to. Your site visitors do not like dead links, and the search engines are offering a service which depends on the visitors being happy when they reach your site. That is the overriding principle of effective link analyzing, and makes it a vital search engine optimization tool to increase your visitor satisfaction.