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How the sync works:

Game objects such as shaman items should be on the same spot on the screen for every player. This is necessary to enable the same gameplay experience for everybody. Therefore one player is chosen (people usually call this player the sync) to keep track of all objects' positions & velocities throughout the current round. This means that this player's game recognizes players interacting with those objects (for example pushing a box around). All objects' positions & velocities are sent every 500ms from this player to all other players in the room. For this player all objects move smoothly around, without lagging. For all other players the objects "jump" into their new position & velocities once they received the updated information from the sync.

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Why the sync is needed:

Players have lag (depending on their ping). If the game would not synchronize the objects every 500ms among all players, the objects would be at different positions for each player, resulting in a different gameplay experience for everybody. For example player A would see a box close to a high ledge so they can jump from the box to the ledge, but for player B the box is still far away from the ledge so they cannot jump there. This would be unfair and chaotic. The sync is taking care that the box is at the same position for everybody. Since the objects are updated only every 500ms and the sync player might have a bad internet connection the objects can still lag, there is no way to solve this problem.

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How the sync is chosen:

Longest cumulative connection time is not the connection time of your current session. It is also not the time since you made your account. It is the time that your profile has been connected to the server in total. Imagine you have a stopwatch that you start when you first log in to the game. If you log out you pause the stopwatch. When you log in again, you start the stopwatch and it keeps going from the time when you logged out. This time is the cumulative connection time. (It's comparable to the steam function which tells you how long you played a certain game.)
The flowchart says blue shaman because of the possibility of double shaman maps.