A few years ago, Google started running encrypted searches on its flagship search engine site, allowing the users to search using https -- the web security system which sometimes associates with online banking and other stuff.
Known as https://, this feature encrypts the data sent between a user's browser and Google's servers, which makes it impossible for someone in the middle to read it.
When you search with https://, the secured channel helps protect your search terms and your search results pages from being intercepted by a third party on your network.
When some one searches through Google encrypted search, it is not archived in history, which means it won't be appearing in the autofill while you search another time.
These days, most of the sites on the first page of Google are HTTPS, since Google prefers them over the other. They want to ensure the best user experience for their customers. In case if your site isn't secure, it could be getting outranked by similar sites.
Implementing this might give the customer a sense of peace and satisfaction that the website they are going can be trusted and their information is safe.