Referencing this tweet: https://twitter.com/notericmin/status/1263219833925066752?s=21
Was this like a break for internal discussion or something? Or were they literally just sitting across from each other for two hours as the transcript would imply?
Yes, it was two hours of staring.
...ok, even if we weren't on AskHistorians, I'm sure you'd want a little more detail. Here goes:
The Armistice Agreement for the Restoration of the South Korean State is infamous for not only taking over two years to negotiate but in the end not even being a permanent peace treaty between nations. No nation was a signatory.
Psychological warfare was initiated at the very start. When negotiations began in July of 1951; the NK greeted the UN with vehicles with white flags (trying to indicate the other side was in fact surrendering).
While the US had previously endorsed a line at the 38th parallel, the initial proposal during negotiations was for a demilitarized zone entirely within the North Korean side; this attempt at a show of strength backfired (and according to James Matray, it was the primary reason the negotiations lasted so long).
As things dragged on, other psychological tactics including having armed guards staring at delegates as they arrived, and arranging seats to make sure the NK representative sat in a chair higher up than the UN one.
On August 10th, the UN wanted to discuss the demilitarized zone being based on the existing line (as opposed to the prior proposal). The NK side refused.
Now, for the exact moment from the tweet you linked, which I'll include the transcription of; the two people here are Vice Admiral Turner Joy and General Nam Il.
NK: At present we have nothing more to say.
UN: Neither do we.
(Pause of two hours and ten minutes).
UN: Would you like to say something?
NK: At present I have nothing to say.
Vatcher records that General Nam Il had a snarled face for the two hours.
Sources:
Matray, James I. "Mixed message: the Korean armistice negotiations at Kaesong." Pacific Historical Review 81.2 (2012): 221-244.
McMahon, Robert J and Thomas W Zeiler, eds. Guide to U.S. Foreign Policy: A Diplomatic History. 2 vols. Thousand Oaks, CA: SAGE Publications, Inc., 2012.
Vatcher, William Henry. Panmunjom: The Story of the Korean Military Armistice Negotiations. Greenwood Press, 1958.