It just seems odd to have someone under the bed and to have sex above them, and then chat with Dunyazad as she sits under the bed. There is no explanation for this. What was the significance of this action?
EDIT: in the context of the story, Shahrazad asks the king if she can say good-bye to her sister, Dunyazad before she is put to death. The King agrees, Dunyazad i summoned, the sisters say good-bye, Dunyazad goes under the bed, the king and Shahrazad have sex, then Dunyazad asks if Shahrazad will tell them a story. The king agrees, and Shahrazad tells both of them a story while Dunyazad is under the bed. Repeat this every night for 1001 nights.
I suspect that "under the bed" is a later addition by some prudish translator. You don't mention which version of Thousand and One Nights you refer to, so it's hard to tell.
Briefly, Thousand and One Nights is not a single book. It exists in several different versions, each of which have one or more translations. The collection that eventually became the book was originally compiled in Sassanid Persia, in the 4th-6th century AD. It may derive from even older sources, since many of these earliest stories are taken directly from the Indian Panchatantra in Sanskrit.
After the Sassanid Empire ended and the Islamic conquest of Persia, these stories were adopted by the Arabs. There are different Arabic versions of the Thousand and One Nights, belonging to one of two main traditions - the Egyptian and the Syrian, each with somewhat different stories and different styles. The Arabs continued to add to these stories for a thousand years. Modern translations can be quite different from each other based on which version they're based on.
None of the versions I've read mention Dunyazad sitting under the bed. They all mention that Dunyazad was present in the bedchamber, but don't say a word about where she was sitting. They imply that Dunyazad was in the bed too, not under it.
For example, here's the first night according to the John Payne translation (which comes from the Egyptian version):
Scheherazade rejoiced with an exceeding joy and made ready all that she needed, and said to her sister Dunyazad, "O my sister, note well what I shall enjoin thee. When I go up to the Sultan, I will send after thee, and when thou comest to me and seest that the King has done his will of me, do thou say to me, 'O my sister, an thou be not asleep, tell us some of thy delightful stories, to pass away the watches of this our night.' Do this and (God willing) it shall be the means of my deliverance and of the ridding of the folk of this calamity, and by it I will turn the King from his custom."
Dunyazad answered, "It is well." And the Vizier carried Scheherazade to the King, who took her to his bed and fell to toying with her. But she wept, and he said to her, "Why dost thou weep?" "O king of the age," answered she, "I have a young sister and I desire to take leave of her this night and that she may take leave of me before the morning." So he sent for Dunyazad, and she waited till the Sultan had done his desire of her sister and they were all three awake, when she coughed and said, "O my sister, an thou be not asleep, tell us one of thy pleasant stories, to beguile the watches of our night, and I will take leave of thee before the morning."
"With all my heart," answered Scheherazade, "if the good king give me leave." The King being wakeful, was pleased to hear a story and said, "Tell on." Whereat she rejoiced greatly and said, "It is related, O august king, that ...
There's no mention of sitting under any bed. It's pretty straightforward: Shahriyar takes Scheherazade to bed and is fondling her, Scheherazade starts crying, Shahriyar asks why, she says she wants to say goodbye to her sister. So Shahriyar sends for Dunyazad who arrives and waits until the king finishes making love to her sister Scheherazade, then prompts Scheherazade to tell a story.
She's not hiding anywhere, she's not under the bed. Scheherazade doesn't say "when you hear that we've finished making love then ask me to tell a story", she says "when thou comest to me and seest that the King has done his will of me, do thou say to me ..".
Now you could argue that maybe Dunyazad was standing at the side of the bed, politely waiting for them to finish, but the story specifically says "and she waited till the Sultan had done his desire of her sister and they were all three awake, when she coughed and said ..". Why mention that all 3 were awake - surely if Dunyazad was waiting for the couple to finish, she'd naturally be awake. It seems likelier that she was in bed too, and the story is telling us that they'd finished but were still awake.
Remember, these stories are fictional (so far as we know), so they're not talking about any real prince or princess. They reflect the customs of the times. The core of these stories and the framing device (Scheherazade telling one story each night) goes back to Sassanid Persia. Those are the times they reflect. If you're willing to accept a plot device that says the king married a new virgin every day and executed her the next morning, is it really stretching credulity that much farther to acknowledge that maybe these people didn't mind having two women in bed at the same time?
Like I said, I don't know which particular translation this "under the bed" thing comes from, but it sounds to me very much like once the story was adopted by the Arabs and became part of the Islamic world, some scribe decided that it wouldn't do to have two sisters in the same bed with the king, so he decided to move one sister to under the bed.
Follow up question: is this an acknowledgement that voyeurism may occur in private, but it is publicly condemned?