The whole universe is explained by advaita along the lines of a classic case of illusion, such as the illusion of the snake in the rope. What is seen under the influence of illusion is the snake which has been superimposed on the rope. The whole illusion is the result of two powers of avidyA (aj~nAna) called the AvaraNa-shakti and the vikShepa-shakti.
Two phenomena happen in this illusion. First, the reality, the rope, is obscured or concealed. This is done by the AvaraNa-shakti. Second, the illusory object, the snake, is projected by the vikShepa-shakti. The AvaraNa shakti is called the power of concealment and the vikShepa-shakti the power of projection.
Nevertheless, the rope is the substratum of the illusion. In the illusion process, the rope gets apparently transformed into the snake. Similar is the case of the brahman-universe illusion. It is only admitted by advaita that the universe (world) is an apparent transformation of the substratum Brahman, not a real transformation. According to realist schools such as the one by RAmAnuja, Brahman is really transformed into the universe. Such a real transformation is technically called "vikAra." But, according to advaita, the transformation of Brahman into the world is just apparent. Such an apparent transformation is called "vivarta." So you may say that advaita upholds vivartavAda.
In the shvetAshvatara upaniShad, it is stated that the power of God is hidden within its own effects, (devAtmashaktiM svaguNairnigUDhAM). And God is nothing but Brahman with the limiting adjunct of mAyA which has a preponderance of shuddha-sattva, pure sattva-guNa. The power by which the universe is created is the vikShepa shakti. So the universe is a result of the vikShepa shakti of avidyA. Considered from one viewpoint, Brahman is the efficient cause of the world. From the viewpoint of its upAdhi (limiting adjunct), Brahman is also the material cause of the world.
Q: But if this empirical world is an illusion, then what am I who am part of this empirical world? I'm an illusion too!
A: Since we are within the context of shAkta upaniShad-s, let me make use of the same. In the sarasvatI-rahasya-upaniShad, it is stated that five factors are observed here in the illusory world. These are 1) asti, interpreted as Existence, Sat, 2) bhAti, interpreted as Consciousness, Chit, 3) priyaM, interpreted as Bliss, Ananda, 4) nAma, name, and 5) rUpa, form.
Of these five factors, the upaniShad states, the first three, i.e. sat-chit-Ananda pertain to Brahman. The last two, nAma-rUpa, pertain to the world.
Coming back to your question/doubt that you are yourself an illusion, the aspect of you that is sat-chit-Ananda is not an illusion; it is Brahman. The aspect of you that has to do with nAma-rUpa is an illusion. So long as you think, "I exist, I am Conscious, I am dear (to myself)", you are on the right track. But when you think, "I am such and such a person with such and such a body, such and such possessions, etc.", you are not on the right track. The gist is that when you start imposing limits on your sat-chit-Ananda aspect, then you are indulging in illusion.
So your contention that you are part of the world and are therefore an illusion is not entirely correct. The nAma-rUpa aspect of you that you share with the world is an illusion, but not the sat-chit-Ananda aspect which can never be negated, so Vedanta tells us.
Q: Then the empirical world that I experience is a result of my ignorance?
Q: But if I myself am an illusion, Isn't the ignorance an illusion too?
A: I have answered above that the sat-chit-Ananda aspect of you is not an illusion. The ignorance creates the illusion. The ignorance as well as its effect, the world, cannot be classified as real or unreal. They are not real because they disappear with the dawn of knowledge. They are not unreal because they are felt during the illusion phase.
Q: Doesn’t this mean that j~nAna, sAdhana and whatever effort that I might take towards salvation, illusions too?
A: As I said the sat-chit-Ananda aspect of you is the only Reality. All sAdhana that is performed, effort, j~nAna are all geared towards removing the illusory aspect of you. Once the illusory aspect of you is removed, they too disappear with the illusion. They are like the illusory tiger in the dream that causes one to wake up. Though the tiger was illusory, the effect was real. Similar is the role of jnAna, Guru, God, and shruti.
Q: I am such and such a person with such and such a body, such and such possessions, etc. Who exactly is the "I"? Is it the name-form or the sat-chit-Ananda?
You say that when you start imposing limits on your sat-chit-Ananda aspect, then you are indulging in illusion. Again who's the "you"? The name-form or the sat-chit-Ananda?
A: The "you" I was referring to is not the sat-chit-Ananda aspect of "you" because this is Brahman. Brahman cannot engage itself in any activity, not even the subtlest one!
The "you" that must discover the real "you" by means of Atma-vichAra (deliberation on the Self under the guidance of a proper guru) is what is conventionally called the jIva, or the individual. So when I say, "you should deliberate on the Self", I mean you as you are conventionally accepted in the vyAvahArika world, as a person, with a name, form, and other limitations (upAdhi-s) imposed on the real "you."
So this individual or jIva should conduct an inquiry into itself. In the ultimate, pAramArthika sense, even this inquiry is in the realm of mAyA or illusion. But, as with the analogy of the tiger in the dream, the inquiry or deliberation is part of the illusion. It is only when the tiger comes alive and roars that we wake up!
Q: If "I" am actually the sat-chit-Ananda, which is the Eternal Absolute, how can my name and form and the illusions affect me? If they affect me doesn't it imply change, to which "I" being the Eternal Absolute am impervious?
A: This is exactly what is required to be realized to the core of one's being. As you have stated it, it is a just a collection of words which can be understood at an intellectual level. But for illusion to disappear, we should go beyond intellectual understanding and realize that we are just sat-chit-Ananda, and cannot be affected by illusion.
I must also point out that this realization is not a mere "mind-game" or self-hypnosis into believing that one is sat-chit-Ananda. You hear statements to the effect that "if you think you are liberated you are; if you think you are bound you are", and so on. To this I will add one more pithy saying as a rejoinder: "If something sounds too good to be true, it is." :-) (This is the rule followed by the intelligent American consumer, and put into practice in throwing away those letters from Publishers Clearing House even if they can potentially make one a millionaire several times over!)
There is a definite discipline to be followed, as established by our AchArya-s. Only by following the discipline, will we ever have a chance for Self-realization.
But until then, the answer to the question as to how illusion can affect us, is avidyA or ignorance. It is only when avidyA and its effect are removed, do we realize ourselves to be sat-chit-Ananda.
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