One of the ways that Google built its business and brand in the early days was by powering search on third party sites. Lots of newspaper publishers for example offered Google site and Web search on their sites. But it was pretty extensive beyond that and drove considerable volume over time.
Now Google is offering custom search to mobile sites. This is a syndication play that may help drive additional Google mobile search volumes, which have grown 30% Q over Q. Ads and monetization are also a part of it (and part of the appeal for some publishers). See, for example, this search for "sushi":