Based on my knowledge of chess tournaments from this time (binge watching The Queen’s Gambit), they have a guy whose job it is to update a large representation of the board on the wall as the game is taking place, so the audience can see

From my experience with the queens gambit, they would have giant boards that people update in between turns.

The moves were announced and I believe they were following along with chess-specific score/move sheets in their hands. I wouldn't be surprised if the move announcements were all that would be necessary, though for a bunch of chess fanatics.

I saw a documentary that featured a dramatic reenactment of a scene like this and that's how it went 🤷

At the time of this match, I do believe they had overhead projectors (the kind that magnify whatever is placed on their surface). So they might have had a projector-specific (with the necessary transparency) chessboard projected behind/above the camera also showing the moves.

Another way I've seen it done is with magnetic pieces on a giant blackboard-like surface that had the squares drawn/painted on. There would be a person on each side with a pole that could move the pieces. One person would be assigned to white, the other to black.

If everyone is sitting up high like that a simple extra large chessboard on the floor nearby would suffice as well. They could even place it down at an angle for the audience to view. I don't know if they did that but I was thinking about it, "how would I do it without modern technology?" That's probably how I'd do it 🤷