The Search element that appears to the right of the Main Menu is a Menu item, and which includes a Search module. To recreate this as it appears in the demo, follow these steps:
Navigate to Admin > Menus > Main Menu and create a System Links > Text Separator menu item.
Under Dropdown Menu Options, add hidden-phone menu-item-search in the Custom CSS Class field.
Set the Menu Title to Search.
Under Link Type, toggle the Add Menu Title option to On.
Set the Status to Published and assign its Menu Location to Main Menu.
In Dropdown Menu Options set Child Item Type to Modules and select Menu Search for Child Modules. You can do this step once you create Menu Search module as per the instructions given below.
Now, you will need to create the Search module to place in the menu-search module position. This module will appear when the search icon is selected in the menu.