Assuming you are talking about Live Analytics in particular:
One of the most common reasons this was requested was for events and product/feature launches. Some of our customers would launch a new product only to find out a day or two later that users were using different terminology or looking in the wrong places when searching for documentation or support. We are seeing content teams be able to adjust their copy in real time based on the search activity, which is pretty powerful.
There is also a lot of value in seeing a user's search activity while they are still on your website. We use Olark, so if someone isn't engaging with results/etc, we can guess at his/her intent and send them a message asking if they need help.
