The immediate reason why Rudolf chooses to open the green door is because a man distributing advertising cards has given him two cards upon which are printed the words "The Green Door." Rudolf is particularly fascinated by these two cards because all the other cards the man is distributing contain advertisements for a dentist. Rudolf feels that not once, but twice, he has been chosen by "the arch sprite Adventure"; a sprite is a legendary creature such as an elf or fairy.
On a more general level, Rudolf is a person who seeks adventure wherever he can find it. The narrator of "The Green Door" praises Rudolf as "a true adventurer":
Few were the evenings on which he did not go forth from his hall bedchamber in search of the
unexpected and the egregious.