My understanding of this method (based on comments from Google that seem to be intentionally non-specific) is as follows:
Google knows and acknowledges that although the
SEO community stays on top of every tidbit of information coming from the Googleplex, the average webmaster/small business person is relatively oblivious to the hows and whys of rankings. As a result, most systems err on the side of caution - punish a site, not a server; punish a page, not a site; punish a link, not a page.
As the Googlebot reads a page, it breaks the content down, trying to figure out what the different parts of the page are. For instance, the bot will look for a footer so that can be ignored from the keyword calculations. Most sites have visual separations between different types of content - think of your own site, a visitor can immediately spot "this is navigation, this is the logo, these are ads, this is the footer". These distinctions are carried over into the code. Googlebot tries to find the same elements to determine how to rank different things. Wording that is found in the header, footer and navigation are discounted relative to the value of the content. In the same way, links that are separated from the content are discounted (or completely ignored) when the value of outbound links is calculated. In most cases, the bot will "ignore" sections of the page if it detects possible paid links. The idea that links within content are worth more because the surrounding text imparts more relevance has already been discussed, this idea simply expands on that concept.
Nofollow enters the equation because if the spider finds these links greatly outnumbering the editorial links, or in some other way these paid links trigger a red flag for the spider, it may decide that the purpose of the page is to spam the index. In that case the page could be penalized by having all pagerank removed, or even be dropped from the index. If this happens to enough pages on the site, then the site would risk a penalty. Finally, if enough sites on the server get penalized, the server itself may be penalized. Nofollow simply gives the webmaster a way to avoid the risk of too many false positives.
Bear in mind, paid direct links were created
specifically for the purpose of gaming search engines. If you bought a link on a web site for any reason other than to increase your IBLs, they would not be direct links. If you sold links for any purpose other than to pass on page rank, you would not use direct links.