How do I get a copy of the searched text inside my search box once search operation results show.

The following code placed on your template page should work: 
<!--SEARCH-GENERAL <input type="text" name="search" value="$search"> -->
If you want to avoid the search options, use 
<!--SEARCH-GENERAL <input type="text" name="search" value="$query"> -->

The $query field was added in Alkaline 1.6. Use $post.search with older versions, which contains the proper search query in 90% of the cases.