Cuddly Teddy Fish was the first person to solve. Hir explanation was so good and/or I am so lazy, that I'm just going to copy it.
If there is only one monk with red eyes, then he sees all the others are brown-eyed, so he must be the red-eyed one. He kills himself the first night.What I always found mind-blowing about this puzzle is that if there are multiple red-eyed monks, then they already know from the beginning that there is at least one red-eyed monk. The tourist didn't really tell them anything new.
If there are two monks with red eyes, then each sees one monk with red eyes and reasons that if this other monk is the only monk with red eyes, he will kill himself the first night. Neither monk kills himself the first night, so they each reason that they must have red eyes too. Both kill themselves the second night.
If there are three, each expects the other two to commit suicide the second night. This doesn't happen, so each deducts that he must be a third, and the suicides happen the third night. Extends to four, five, etc.
If the suicides happened n midnights after the tourist's remark, then there are n monks with red eyes.
But the tourist did convey a key piece of information. Because of the tourist, all the monks know that all the monks know that all the monks know that all the monks know [etc. etc.] that there is at least one red-eyed monk.