No one has any hesitation, obviously, in taking the dream world as an illusion; for, when they wake up to this familiar world, the dream world disappears. But all of us find it hard to believe that this familiar world, which we all actually perceive and experience, is an illusion. If I jump down from an upper floor, I will certainly be fatally injured. This difficulty is because we are all part of this world and are attached to it in some way.
But a spiritual aspirant may ask, "Is there a higher state to which I can wake up, so that this waking world will disappear, just like the dream world?"
The answer is a resounding "yes." What that higher state is no one can precisely describe. But Sri Shankaracharya was an intellectual and spiritual prodigy. He could experience that sublime, transcendental state (turiya, wherein the jiva is in a state of complete identification with Brahman), just like the Upanishadic seers. Thus, the great Acharya could confirm and authoritatively summarise the vision of the ancient seers of Sanatana Dharma — the truth of Advaita. Before he left his mortal coil, he firmly established this philosophy by his masterly commentaries on the prasthanatraya (the three basic texts on Vedanta, viz., Upanishads, Bhagavad Gita and the Brahma Sutras).