For many website owners the change to secure search (SSL) on organic search has made a huge difference to how you can understand visit intent. Being able to understand visitor engagement segmented by keyword provided insight into visitor needs. Armed with the knowledge of searcher needs a copy writer could change the site to be engage visitors more.
If you are serious about understanding the intent of your visitors. The thing to remember is that there are no short cuts in your journey to understand your visitors. Google webmaster tools can give you a lot of insight into what keywords are working for you. Given an understanding of your visitors intent in general you may have to get serious to understand then...
Armed with a list of keywords generating traffic from webmaster tools you can start planning a AdWords campaign to get your assumptions. Given your site wide conversion rate pick keywords with a good chance of having a positive return. For those keywords run a mini campaign to find of what works. Ideally use your page meta data to create copy as similar as possible to your page's copy. The goal is to buy relevant visits and learn what keyword and copy best attracts relevant visitors for your page.