What does rule 538 mean - specifically, when can players not respond with reactions before a chain resolves?

Ruling: Rule 538 means you cannot play Reactions when a permanent card is played (usually from hand, but also from hidden). You skip priority and go straight to resolving the chain. However, activated and triggered abilities of permanents CAN still be reacted to, because abilities are not themselves cards.

Nuances:

• Playing a permanent from hidden still counts as playing a permanent card, so no reactions allowed
• Token units created by abilities can still be reacted to because abilities grant priority
• The rule applies specifically to permanent cards being played, not to abilities of permanents