Intuitively, it is a good idea to imagine it like this. It has a big neural network which receives the current board position. Some of the nodes encapsulate ideas like how much material there is, who controls territory, what is attacking what, what skewers and so on through all sorts of configurations. Also positional factors of a zillion different types.
These abstract concepts appear in the network at deeper levels as it trains because they happen to be useful for working out what is a good move. You can think of it as a bit like evolution (only it is a bit more deliberate in the way it adjusts the parameters to try to make the evaluation better).
As an analogy from another area, neural networks are used to classify photos. The first level of the network may identify simple things like whether adjacent pixels are similar. A higher one may identify edges. A still higher one may identify face shaped areas, another something that might be an eye. A higher node still may provide a probability that a face is your face, out of a list of faces in photographs on which it trained.
AlphaZero has to do the same for chess. In some ways this is easier, in others harder, but the tool in both cases is a large, deep neural network made of very similar dumb units that learn to be useful.
It's worth pointing out that AlphaZero does not have a database of positions. It has a convoluted way of calculating a number when presented with a position (and a list of other numbers for the legal moves). The origin of this convoluted function is its reaction to experience of about 28 billion positions and what they led to.
Then it is all about memory rather than AI.
Ermm...no. I believe he's saying that AlphaZero's convoluted calculations are pretty much similar to, say, tactics trainer positions for a human being. AlphaZero will not traverse thousands of games each time to find one that matches...it has developed an "eye" for positions and will apply its uniquely developed valuations of each factor in the position without the need for database lookups of previous game moves.
In that sense, AlphaZero will be like a unique human player. It played itself for millions of games to "figure out" chess. If it had played Stockfish for millions of games instead, its understanding of chess would be uniquely different, and if it only played humans worldwide online or something, its understanding of chess would be unique in a different way.
It is the right way to go, leaving behind all human history and the faulty assumptions we have concerning best play, which are also built into our engines. It's just too bad they jumped the gun on announcing dominance over the best engine under dubious circumstances.