Any user that has the right to edit a certain Silk site, will see the "New Page" button. Clicking it will reveal the "New Page" dialog and will enable you to add a page to your Silk site.
Make sure to enter both a page name and a page category. The category is optional, but very useful if you want to use Silk features such as inserting widgets or running queries in explore mode.
If you enter text in the search bar before hitting the "New Page" button, this text will appear in the Page name field in the dialog.
Instead of using the button, you can also enter the desired page name in the search bar, and then choose the "Create new page" option in the menu that appears.