Flex fins are just what they sound like - offering more flex that a traditional longboard fin. The added flex makes the board feel a bit looser and adds a bit more liveliness/springiness to the board which can bring a whole new flow to your surfing.
Flex fins tend to distort as they are turning through the water, allowing the board to go faster through the turn. This also lets the rider use more of their rail line.
There are different levels of flex that you can get in these fins, and the amount of flex will impact the amount of drive a fin will give when you really lean on it. If you want more drive out of our your turns - a fin with moderate flex in the tip may be the trick. Of course everyone's style is different, so its up to you to experiment!