Given a board configuration, many expert chess players are able to reproduce the moves which produce that configuration. However, of the 10^40+ board positions, grandmasters would be hard-pressed to reproduce the vast majority of board states. Why is that? Well, that's because most of them involve obviously bad moves, like developing your king by itself to the center of the board, or sacrificing all your pieces willy-nilly. The games which produce these board states are not interesting. A much more difficult question is: "How many board states are likely?" What if that number is just 10^10 or so?
Computer science has one powerful optimization trick. When performing an expensive computation which always has the same result for a particular input, the programmer can instruct the program to simply remember the answer. This technique is called "memoization". Then, the next time the computation is requested, the program "cheats" by returning the memoized value instead of wasting a bunch of effort computing something exactly the same way it did before.
Humans also have one powerful optimization trick, called the Hebbian Learning Rule, often summarized as: "Neurons which fire together, wire together." It is overly simple to say that this explains all human learning, because it doesn't. But it is, in a sense, the computationally simplest and most abstract way to capture what the memoization technique does: if a pattern of neural firing consistently produces the same result, let's take a shortcut and not do all the work we did the first time.
Now, let's compare those tricks to what Stockfish does: brute force search of the game tree. Obviously, Stockfish is smarter than that. It can use an opening book, and endgame database, and it can prune branches of the game tree that are trivially silly. But there's one thing that Stockfish cannot do: it cannot say: "Wait a second! I've been here before. What did I do the last time I was here?" Stockfish has no memory (across games). Of course, this makes it useful for evaluating positions, because Stockfish's answer does not depend on which games it has already played. It will give you the same answer every time you evaluate the position, even if it plays a game in which it discovers that its evaluation was inaccurate.
So now we begin to see why AlphaZero can evaluate 1000x fewer positions, and still wipe the floor clean with Stockfish: AlphaZero does the simplest thing an intelligent system can do--it remembers the past. You cannot ask what the AlphaZero evaluation of a position is, because that is a meaningless question. Each instance of AlphaZero is unique, shaped by its history of games played. Instead of saying: "I'm going to pretend like I've never seen this board state before, and blindly search the game tree" it says: "Well, I've seen this board state 237 times, and I already know that 3 moves are significantly better than all the others I evaluated" (it doesn't actually know how many times it's seen the board state, except implicitly).
Incidentally, this one simple trick is also why humans are able to evaluate tens of thousands of moves fewer than computers, and still play a respectable game. A grandmaster is not a human who can evaluate 60,000 positions per second. A grandmaster is a human who can remember 60,000 board positions and strong lines of attack and defense from the most important ones.
AlphaZero plays more human chess than any program before it, because it actually learns chess. In a way, it probably understands chess better than any entity in the universe. Its primary shortcoming is a lack of communication ability. If it had language, and could describe its reasoning, we would likely be awed by its analyses. And yet, we would quite possibly be surprised how human-like those analyses turn out to be, while noting that there is still an element of alien sensibility borne from the ability to evaluate tens of thousands of moves per second.
The fact that AlphaZero runs on transistors which can switch a billion times per second, compared to the measly 3-5 times per second for the typical human interneuron suggests that in a sense, AlphaZero is much like a hyper-accelerated human. But to get back to your original question: human grandmasters can look into the vast space of future moves. They do so by remembering which of those moves are better than others, rather than computing them fresh every time. Of course, play is a combination of memory and tree search, so humans, like AlphaZero are also exploring new lines of play every time. But they have to be more economical in their search, and it shows.