Hiya,
Inside the "search_the_internet" template, exists the HTML form... the opening FORM tag has as one of it's parameters "target="_blank", this is what causes the searches to open up in a new page.
The reason being, people may be posting a thread, a reply, doing a PM, or simply just want/need to stay in the same screen, but they need to search somthing at the same time.
If you just wanted it to load into the same browser window, just for your forums search, then you'd need to take out the "target="_blank" paremeter from the FORM tag, and then inside the 'websearch.php' script, we'd need to add some JavaScript to pop-up a new window, if the search wasn't for your forum.
I don't like adding JavaScript pop-ups, as they can be un-reliable. But that would be the solution.
The main consideration would be, visitors needing to stay in the same screen, and do a search at the same time... you'd overwrite what they were doing.
It's upto yourself though.