The Arcade Dungeon is accessed by talking to the Clothier.
Once you enter the dungeon, you are able to gain points for defeating enemies, both your current score and your best score will be tracked.
For every dungeon monster you kill in the dungeon, your score will go up, but monsters will become tougher and tougher.
If you exit the dungeon with a score greater than 0, you will be given coins and exp rewards based on your score. Passing the high score will double your reward, and having an especially high score will make NPCs sell some new items.
Note that this system does not affect the dropping of the custom item Dead Wood. It can still be acquired by defeating Dungeon mobs.