This is the com_search of joomla. You can override the components files, and edit them in order to remove what you do not need.
To override the files, just copy the folder from components/com_search/views/search and paste it in templates/your_template/html/com_search/. Then you will have to move the files default.php, etc. from the subfolder tmpl to the search folder. You will have to also erase the files metadata.xml and view.html.php. All these changes have to be made in your template files and not the core files. Once done your file structure should be like this:
html
-com_search
--search
---default_error.php
---default_form.php
---default_results.php
---default.php
---index.html
-index.html
All of these in case the override does not exist in your template already. OK?
Now, to change the part you are referring to you will have to edit the default_form.php file.