Google Plus added new small feature that has been added on organic search results -- "a thank you link added on organic search listing next to the Google+ users who +1′ed a result," informs State of Search and charliesaidthat (SEL).
The feature is located next to a "your friends X,Y,Z and x-number of people +1′d this" text it now says "Thank them". And you can "thank them" on Google+ off course.
This functionality is currently only showing intermittently at the moment, when logged in as a Google+ user.
In order for the "Thank" link to show, the link in the results had to be +1′ed from a circled user.
When you click on the "Thank them" link an overlay opens over your search results giving you the opportunity to thank people on Google+. If a searcher did choose to thank someone for the +1 they were prompted to thank directly on Google+. The message automatically contains the original +1er on the post:
This limited visibility post is shown directly on the wall of the user who is doing the thanking. If you are looking to keep your appreciation of the +1 private, just make sure you don't add in any other circles to the message: