We Suspect You of a Crime
Tell Us What You Violated
Here is a set of step-by-step directions from SearchEngineWatch on what to do when Google tells you they don't think the links to your site are legitimate. To fend off death by Google snubbing, a site needs to act diligently to ensure all the incoming links from other sites not only are legit, but appear to be legit.
Backlinks are those links to your site out their on the web, the vast majority of which you don't control. When Google sends you a letter indicating they suspect you are controlling some of those links, it means that they expect you to act, or the likely will not continue to serve up your site as a suggested link in organic search results at the same rate.
Google's Matt Cutts put it this way: "So while the site's overall rankings might not drop directly, likewise the site might not be able to rank for some phrases. I wouldn't classify these messages as purely advisory or something to be ignored, or only for innocent sites."
Cutts normally isn't that poor a writer. That phrasing is in keeping with Google's deliberate efforts to ensure that their policies and guidelines remain obscured enough to be difficult to game. (Or, for the cynics, obscured enough so that they can't be called out when they violate their own guidelines.)
Since Google is a firm staffed mostly by robots, a webmaster trying to deal with this situation tends to have one and only one chance to talk to Google humans, convincing them that any offending links have been removed.
Of course, since those links exist on someone else's site, this is an involved process of communicating (pleading) with many other webmasters, and as the article points out, documenting these conversations as further evidence of your site's good faith.
Some search engine optimization professionals disbelieve the bad effects of spammy links, although their numbers seem to be fewer and fewer as Google has been more vocal on the issue.