Unsatisfactory is to describe an object or situation. Unsatisfied and dissatisfied are to describe emotions. How are the two emotions different? Dissatisfied means you are not happy with the quality/performance/ability of something. Unsatisfied is often used to show when we haven't had enough of something. Let's use a meal to show the difference. If the meal is bad, you would say I am dissatisfied by the food. If it was too small and you are not full you are unsatisfied. On a practical level, dissatisfied is much more common.

They are basically all adjectives, so that's why they are a bit confusing. However, here is an easy way to understand them: Unsatisfied is being used to talk about a person or a need which is not fulfilled: She remains unsatisfied with her result. Meaning she expected more. Dissatisfied is being used to express that you are not happy: The dissatisfied tiger went back to its den. Meaning the tiger was not happy. Unsatisfactory is being used when something is done in a poor quality: The unsatisfactory sculpture was destroyed. Meaning the sculpture was not good enough. Does that make sense?