Worthiness is a quality of being suitable or having some kind of value. You can demonstrate your worthiness as a petsitter by showing up on time and taking good care of your neighbor's cats.
When something (or someone) either merits your respect and admiration, or is perfectly suited for the task at hand, you can talk about its worthiness. When you vote for mayor, you probably choose the candidate whose worthiness to lead is most clear. Or you may find yourself in a philosophical conversation with a friend about the worthiness of various professions.