The Woodcutter and his wife live in a run down shack. As the story goes, the Woodcutter's wife is ill and he fears she may die. They have no food and things are not looking good for the couple.
During the game, you can try and help the Woodcutter by giving him various items. The only item that will really help him is a bowl of stew. If you help the Woodcutter out by giving him this, he will give you his fiddle that is sitting on the table. The fiddle is useful in the Leprechaun Cave.
The other items in the images, such as the axe, water pump, vase and so on, are not items that can be gathered.
If you look at the Woodcutter, you will get a small paragraph about how he has been unable to earn a decent living ever scene the woods became too dangerous to venture into. It can be assumed the Witch had something to do with this as the area around her house is pretty much dead. So, although it's not mentioned in the game, getting rid of the Witch may help out the Woodcutter as well.