Each page within the search app has its own template to display that page's content.
Where to save template files
Template files will go into the search folder inside the templates folder.
If your theme was called "Custom" then this would be an example folder structure:
Search app pages and their templates
Below is the list of search app pages and the name of the template file that you should create.
The search page is used to show the search form.
Template name: search.twig
The search results page can show the paginated results of a search.
Template name: search-results.twig