The phrase implies that it is nice to have guests at your home as long as they stay with you for a short period of time. After a while, they will become an inconvenience or a nuisance no matter how pleasant they initially were. It compares guests to fish because the food can have a pleasant taste if fresh but will become rotten as time goes by.
Marginal cost is the increase of total cost that comes with making one more unit of an item. For example, if you wanted to bake a cake, and you had no oven, you would need to buy the oven plus the ingredients for the cake. However, if you wanted to bake a second cake, you would only need to buy the ingredients, as you already have an oven.
On the other hand, marginal utility is the usefulness that a second unit of an item has compared to the first unit of it. For example, if you were dying of thirst, the first glass of water would be very important to you. However, each subsequent glass would become less and less satisfactory.
In this example, the marginal cost of a guest staying an extra day keeps rising (you are more annoyed each day, so each new day becomes more of a burden on you), while the marginal utility keep decreasing (each day with visitors becomes less satisfactory).