Let’s consider the board on page ten. It has the following as the first move:
☗７六歩
Since this is the side that goes first, the reading begins 「先手」. (For the second player, it’s 後手).
Next, you read off the destination: なな、ろく
After is the piece: 歩
All together, ☗７六歩 is 「先手７六歩」
Movement can be added after that, to help resolve ambiguity (if there are two possible pieces that could have made the move). You’ll see 打 for a piece dropped (placed onto the board), and 上 when a piece moved upward.
Promotion information may be added to the end as well.
To see it in action: