The floor mat costs 350 coins. It requires you to unlock the floor mats skill before being able to use it.
The floor mat is only used to minimize the amount of dirt your customers bring into the shoppe. It works passively, meaning once you place it on the floor you no longer have to interact with it. That being said, you also cannot repair floor mats. As customers step over the mats, they will lose durability and will eventually fall apart.
If you're having trouble with the amount of dirt being produced by your customers, you should try completing the Skimpy Leather quest as it unlocks the Leather Statue which reduces customer dirt production by 50%.