The main cause of bondage is identifying Self with body and mind, both of which are not eternal. The body is constantly changing. We assume doer ship through the mind - which we identify as our Self - and any action performed by thought, word and deed is wrongly associated with Self. But the Self is non-doer; it is just a witness. Only the ignorant consider the Self as doer. A person who is aware of this truth is free from bondage of the subtle body which is the mind and intellect concept. Such a person is able to evolve to higher realms.
The sentence “I am not the doer, Providence alone is the doer” means that my wisdom and the thinking capacity that is in me would not have been possible if the One present in all of us is not guiding my thoughts. I am not the actor, writer or performer - someone is acting, writing or performing through me; that One is the Cosmic Self, whatever name we might wish to assign to the Power.
It is the knowledge of a Jnani or the emotional feeling of a Bhakta that brings out non-doership which annihilates ego. An unchecked ego hinders the flow of thoughts. Tons of praise may come your way from those who appreciate creative work produced by your efforts. And the praise may sometimes make you feel proud. Yet it makes sense to let all the praise reach the One who is guiding you, so that there is no hindrance for further flow of thoughts.
You don't have to imbibe non-doership; it should be as naturally present as hunger is present. Does a lion need to learn how to hunt? Or a deer, how to flee? No, it is just there as a natural instinct. Similarly, having the skill to understand non-doership should come naturally to the one who has become the Drashta or seer; or perhaps who is just a witness. Only the ignorant would say that every event is happening due to them.
For Karya or something to happen, three things are necessary: The person who does the job, parent material and the instrument to do it. Take, for instance, the making of a pot. You have the potter who makes the pot, the parent material is the clay and the instrument to make the pot - the potter's wheel. If the pot (Karya) has to be made, the potter will use clay and the potter's wheel. Here the potter is Abhinna Karana or undifferentiated cause, clay is Upadana Karana or essential cause while the potter's wheel is Nimitta Karana or the instrumental cause. Hence the potter's wheel cannot take credit for the pot. The credit goes to the potter and clay. A potter can choose any potter's wheel he wishes. Similarly, in our lives we are all like the potter's wheel in the hands of the great Potter, the Supreme. Neither are we potter or clay; the Supreme is all three.
What is so special in having an attitude of non-doership? At the gross level it will keep ego in check. At the subtle level, it will not let you become the bhokta, the enjoyer. I am neither karta nor bhokta. This will ensure that one does not return in another life to enjoy or suffer the fruits of past deeds. The attitude of non-doership would put an end to the cycle of birth and death.