The Story shows you how you've been matched with someone. Specifically, it details the traits and attributes from your post you had in common with your match and the ones you missed.
The Story gives you that extra level of comfort and confidence to decide whether or not you want to further engage with your match and start chatting.
If you feel good about it and want to move forward tap on the Like (heart) icon. If you don't feel it tap on the Cross icon.