As far as I know the built in search and statistics function in EPiServer CMS does not provide any assistance in this area. To extract this information you will have to rely on external statistics tools such as Google Analytics or similar.

If you use Google Analytics you should make sure your search form is a real HTML form with method="get" so that the search phrase ends up in the querystring. Then you can use GA's built-in "search page" setting and just input the name of your search phrase parameter.

You could easily create a simple statisticslogger by using log4net.
Create a loggappender (you can find an example in EPiServerLog.config)
Create a class utilizing the apender with one log method.

Log the search strings when processing the search.

Google Webmaster tools including Analytics are terrific, but you could also integrate EPiServer CMS with EPiServer SEO.

Working with Search Engine Optimization

The Editor Advice tab in EPiServer SEO shows the quality of the content on a specific web page, and give you non-technical, easy to understand feedback on the areas you have done well and specific recommendations on how to improve. For example, "Title tag is too long".

By using key phrases instead of single keywords on your website, you can work with the title tag, meta description, headings, such as <H1> and <H2> tags, lead text, image Alt-tag, URLs and hyperlinks, that are all together key areas in establishing high search engine visibility. 

Generating Statistics

The administrator of EPiServer SEO can make settings for prioritized keywords/key phrases for the entire website and provides an automated, updated rankings report for those keywords on major search engines. It is also possible to see which search phrases that have generated visits from search engines to this website during the last 30 days.

As said in the previous comments, there are no support for this. However, even though you are an EPiServer-rookie I can have a solution for you if you want to use search query information on your site and not only for editor statistics. Going through logs is not an option for me, I rather implement functionality that handles this for me.

When the search is triggered, save the query into the DDS (Dynamic Data Store), for example as a search object. Create a method that fetches the most queried words and display them on your page. Pretty easy to implement and can be extended with a lot of interesting features. 

Good luck. :)

/Andreas