Most of the web rather resembles black holes that cannot be seen and from where there is no escape, because they do not have links - neither incoming links nor outgoing links. Incoming links have two functions. Firstly, links produce traffic. If your link is shown on thematically related sites, there is a good chance that visitors to these sites will find your link interesting and want to visit your site too. Secondly, each incoming link is a vote for your site.
After all, the other webmaster has reviewed your site and
found that it is of sufficient quality to be listed on his
site. This vote forms the basis for some search engines that
base their ranking of sites in their results on exactly these
votes. The number and quality of these votes represent what
is referred to as "Link Popularity".
Why exchange links then? Because what goes around, comes around. You provide traffic to your link partners, they provide traffic for you. By engaging in a reciprocal link exchange, you effectively vote for their site, and they vote for your site. Thus, as reciprocal link exchanges of thematically related sites are subjected to human peer review, link popularity is a valuable metric for search engines to make it an important factor in their ranking calculation.
The only true way for a good link exchange campaign is to secure quality links by exchanging links with thematically matching sites, where the links are actually to the benefit of the visitor. This will require human review of all links and is usually achieved by emailing the webmaster of the other site about your link exchange proposal.