To clear every search, click on the "Clear Searches" link located at the top-right of the list.
A pop-up box will ask you to confirm the request. Click the Clear Searches button.
Your search history should now be cleared. Simple!
Note, however, that Facebook doesn't let you turn off its search history feature. That means you'll have to remember to manually clear your searches every now and then.
But is there really a downside to clearing your Facebook search history?
Like Google, Facebook says it uses your past searches to deliver more relevant results. Personally, I haven't found that it makes any real difference.