Last August, Google began displaying a save button on local panels on mobile search results. Now, some are seeing the save button on the desktop version of the local panels as well. Personally, I do not see them yet but both Phillip Barnhart and Sergey Alakov do see them.
Here is a screen shot Phillip Barnhart posted on Google+:
This is what I see:
Sergey Alakov spotted this and notified me of this on Twitter. It is unclear if this is a test or a rollout about to happen.
You would be able to access your saved places in Google Maps.