I’m so glad you asked! This all took me forever to figure out, and it’s still hard to remember, which is why I made the chart.

The black or white shape tells you whose move it is–their name is written in there. It also tells you who is
先手 (black) and who is 後手(white). The 先手 is the person who moves first.
The two numbers tell you the position the pieces was moved TO, according to the numbering shown above. (Which is how the 先手(black) sees it). Since you don’t know where it was moved or dropped* from, you kind of have to figure it out. Drops are not noted if that’s the only way to get that piece in that spot. If there are moves or drops that would look the same, a drop is noted with 打. If two of the same type of pieces can move to the same spot, it will be specified left(左) or right(右), retreat (引) or advance (進)
*Drop = play a captured piece.
Each promoted piece had a new name, so if it’s not promoted, you just keep using the old name. If they decide to promote, you specify 成 (なり)。
The kanji character is the abbreviation for the piece that was moved or dropped. I included the abbreviation in my handy one-page guide to shougi pieces chart.