This can be done either server-side or client-side. The search keywords are determined by looking at the HTTP Referer (sic) header. In JavaScript you can look at document.referrer.
Once you have the referrer, you check to see if it's a search engine results page you know about, and then parse out the search terms.
For example, Google's search results have URLs that look like this:
http://www.google.com/search?hl=en&q=programming+questions
The q query parameter is the search query, so you'd want to pull that out and un-URL-escape it, resulting in:
programming questions
Then you can search for the terms on your page and highlight them as necessary. If you're doing this server side-you'd modify the HTML before sending it to the client. If you're doing it client-side you'd manipulate the DOM.
There are existing libraries that can do this for you, like this one.