SPY x FAMILY - I like that Loid is somehow losing.

Yes, somehow…
Don't mention that Anya is cheating by reading his mind.

…its a rule set? for competitive chess games?
You must really like living up to your name raptor, maybe you should rename to "Under a rock Raptor" because that seems to be how you've lived most your life.
I don't know the full details but from what I understand, the rules with the clock are as follows – each player is given a limited amount of time (example – 30 minutes for each player = 60 minute game) for any and all actions taken between each turn (once you press the button your turn is over and the clock timer is on pause) and this is meant to encourage each player to spend their allotted more efficiently/spend more time being active on the board as to not make each game a slog.
I don't know the full details (someone correct me if you do) but I assume these rule sets are specifically done for competitive tournament chess games with the idea of people playing more then 4 to 5 games at a given day.
As for people playing chess casually I would assume if people enjoy playing in the style of a tournament or simply to spice up the regular play of chess with a "time limit".