Already discussed about this in my article Google New Site Search Feature.
So its confirmed by Google Search within a site: A tale of teleportation.
Google had introduced this feature for the users who sometimes forget the exact url of a website and dont have much knowledge about how to search within a website using google search. User have to type the website name in google .. i.e “sulekha“.
This will be recognized by term “teleporting”.
According to google websites displaying site search box are chosen algorithmically based on metrics that measure how useful the search box is for the users of that website.