Firstly let's go straight to the source page of the Naoki Inaba site:
The rules translator wrote:
Lines are drawn connecting all the cells. [forming a single connected network]
The complete diagram represents how a tournament occurs.
The three different shapes represent different types of participants.
The Kanji character [translating as "superior"] represents the tournament victory.
Black circles represent a match between two participants.
[The participants and victory symbols have lines in one direction. The black circles have lines in 3 directions. Cells with no symbol have lines in 2 directions.]
The number in a shape is the quantity of matches won by that participant.
The winner of each match continues to a later stage of the tournament. There are no matches which result in draws.
[The winner of each match is determined in a fashion similar to rock-paper-scissors. Square beats Circle. Circle beats Triangle. Triangle beats Square.]
This is the image of an example puzzle (with answer):
This is the image of the stage 1 puzzle from that webpage: