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 🤷