To increase your presence as an online authority. More authoritative sites on a subject come up higher in search results.
Very basically we set up a google plus profile and google plus page, then link them to your website. When someone finds your site in a search, you get a head shot and authorship along with the search result.
An interesting excerpt from "The New Digital Age", the upcoming book authored by Google Executive Chairman Eric Schmidt, made some headlines:
"Within search results, information tied to verified online profiles will be ranked higher than content without such verification, which will result in most users naturally clicking on the top (verified) results. The true cost of remaining anonymous, then, might be irrelevance."