If a player pays {1}, Rhystic Cave does not produce any mana, and is not considered to have been tapped for mana. Things which trigger of the land being tapped for mana will not trigger.

Each player (starting with the current player and going in turn order) gets the option to pay when this ability resolves.

The card’s ability has errata so you can’t activate the ability during casting of a spell or activating of an ability. This prevents you from getting into a position where someone paying {1} could stop you from having enough mana to pay for the spell. If you want to use it pay for a spell or ability, you need to use this card before you start the casting or activating.