Well, I don't know how individual programmers who implemented the various search engines would do it, but, having implemented similar functions before, this is the way I would approach it.
- Assume your search process uses a database. The person posing the search provides the search terms, and your program formulates a SQL query based on those terms. One option in SQL is the "COUNT" option, which returns the number of records in the database matching the query. This number would be placed in the "number of matches" part of the page.
- To control pagination, (how many records to display on each page), you have a program variable, say $records_per_page that is either a constant value (everyone sees 20 records per page), or user configurable. You then need to keep track of what "page" you are about to show. and for each page you will display from record ($page_number - 1) * $records_per_page. SQL fetch nexts can be used to get successive records.
- Finally, since we are talking web pages here (are we?), you need to preserve state between sessions. There is lots of information on this site (and others) on how to do that.