The .searchbar-home class is added to the div that contains the searchbar when on the homepage. This allows sites to customize the searchbar differently on the homepage than in search results pages, and other places the search bar appears. For example, adding the following CSS would create a large, Google-style search bar on the homepage only:

.searchbar-home .search-box {
  width: 80%;
  height: 3em;
}

.searchbar-home #search_qtype_label,
.searchbar-home #search_itype_label,
.searchbar-home #search_locg_label {
  display:none;
}