The classical Buddhist analysis proceeds by examining the five aggregates, physical form, feeling, perception, mental formations, and consciousness, one by one, and asking of each in turn whether it could be the permanent, independent self a person instinctively assumes exists behind their experience; in each case the analysis concludes that the aggregate in question is impermanent, subject to change and cessation, and therefore disqualified from being the fixed self that is being sought, since a genuine self, on the assumptions the argument shares with its Vedantic and Samkhya rivals, would have to be permanent and independently existing rather than conditioned and constantly changing.
What remains, on the Buddhist account, is not nothing, and the teaching is explicit that this is not a denial that persons exist conventionally, act, suffer, and can be liberated, but a denial specifically that any fixed, unchanging essence underlies that conventional, functioning person; the aggregates themselves continue to arise and pass away in a causally connected stream, dependent origination describing exactly how that stream is structured, without there being any further, separate self riding along on top of the stream and owning it. This is what allows Buddhism to affirm karma and rebirth, a causal continuity carrying moral consequence forward, while denying that any unchanging soul is what is reborn, a combination that puzzled and continues to puzzle interpreters trying to square continuity without a persisting substance.