I've had a look at Shankaracharya's entry in Wikipedia, and it's not much help in answering the question. If he did
"defeat" Buddhism, i.e., his teaching was instrumental in people giving up whatever version of the Dhamma
was being practiced at the time and turning towards practice based upon his version of Advaita Vedanta
, it doesn't explain how that process took place. It said that he formulated the difference between the two traditions:
He also explained the key difference between Hinduism and Buddhism, stating that Hinduism asserts "Atman (Soul, Self) exists", while Buddhism asserts that there is "no Soul, no Self".
But there isn't any argument as to why the former position might be more acceptable. The foundations of his reasoning seem to be a mixture of reliance on the upanishads, and direct realisation of the truth due to spiritual practices. These are both beyond the realm of comparison and argument. There is some discussion of whether Shankara's teaching was influenced by Mahayana Buddhism, and the similarities between the via negative approaches to both Brahman
. As to the issue of why he thought that the anatta doctrine was wrong, there isn't really anything to go on.
Some scholars even doubt that his teaching had much of an impact upon Hinduism, and that his influence has been exaggerated by later Hindus. If he did teach at a time when Buddhism was declining, then it's difficult to be certain that the two were causally related. And even if they were, there isn't much evidence as to the source of his persuasiveness, and virtually nothing in doctrinal terms. Personal charisma, maybe, or good organisation (he founded monasteries) or a teaching that resonated better for reasons we can no longer access.