The evaluation function of a chess engine, whether instantiated as a neural net or explicit code, is always able to assign a value to any board position. If you give it a board position, even absurd ones that would never occur in a game, it will be able to spit out a number representing how favorable it is to one player or another. Since the number of board positions in chess is unmanageably gigantic, the training can only occur on an infinitesimal sample of the game tree. The engine is not simply recalling previously calculated values of board positions, but is performing calculations based on the arrangement of the pieces. For a non-neural-net example, part of a chess engine's evaluation might be to add up the value of each piece on its side and subtract the total value of the opponent's pieces. Then, one set of parameter that would be adjusted while training would be the value of each piece.
When the engine is untrained, the values assigned to a position might as well be random since the parameters of the evaluation function start out with (usually) random values. The goal of a training phase is to adjust the parameters of the engine so that it assigns high scores to board positions that are probable winning states for the player.
From the paper on AlphaZero (page 3):
The parameters of the deep neural network in AlphaZero are trained by self-play reinforcement learning, starting from randomly initialised parameters. Games are played by selecting moves for both players by MCTS. At the end of the game, the terminal position is scored according to the rules of the game to compute the game outcome: −1 for a loss, 0 for a draw, and +1 for a win. The neural network parameters are updated so as to minimise the error between the predicted outcome and the game outcome, and to maximise the similarity of the policy vector to the search probabilities.
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In summary, during training, AlphaZero played a game against itself. When the game is over, the result of the game and the accuracy of its predictions in how the game would proceed were used to adjust the neural net so that it would be more accurate during the next game. AlphaZero is not keeping a record of every position it has seen, but is adjusting itself so that it can more accurately evaluate any board it sees in the future.