I'll admit that I haven't seen the series, but let me just add that 'sealing a move' was common practice when playing with adjournments. That practice has disappeared, over-the-board games are now always finished the same day, but it used to be possible for games to be adjourned if they weren't decided after 40 or 60 moves. Wikipedia lists two common time controls, the first one which matches the scene:
- 2½ hours per player for the first forty moves, followed by adjournment (a five-hour session)
- Two hours per player for the first forty moves, followed by one hour for the next twenty moves, followed by adjournment (a six-hour session)
Basically, it worked as follows: a player may choose to 'pause' the game by writing down their move, put it in an envelope, seal it and give it to the arbiter. Both players could take the time to analyze the game in great detail (with their second or their coach). The game would then be resumed the next day, or perhaps even later in the tournament, on what was scheduled to be a rest day.
The main benefit of this system would be that the players weren't exhausted from the first hours of the game, so the quality of play would likely increase. The main drawbacks: tournament schedules could become tight, and it was disappointing for the public if you didn't know who won. Imagine a soccer game in a knock-out tournament which (even after extra time) is still equal, and the players decide that they will come back tomorrow to finish it off...