Why does Juliet allude to Phoebus and Phaeton in this soliloquy?

Answer 1

This soliloquy is a powerful expression of physical desire. Juliet has married Romeo but she has not yet slept with him and she is desperate to do it. "I have bought the mansion of a love, but not possessed it." But Romeo is not coming to her bedchamber until the night, so she has to wait all day for him to come. Her allusion is to the myths which portray the sun as a chariot drawn by "fiery-footed steeds" and steered by the charioteer Phaeton. Phoebus is another name for the sun-god. She wants the guy in charge of the sun to whip the horses "to the west, and bring in cloudy night immediately"; in other words she wants the sun to set so it will be night, so Romeo will come so that she can have sex with him.

Answer 2

You realize, of course, that saying "this soliloquy" out of nowhere makes no sense. Fortunately, your question tells me that it is Juliet's soliloquy in Act III Scene 2 which starts "Gallop apace, you fiery-footed steeds towards Phoebus' lodging; such a waggoner as Phaethon would whip you to the west." because there are the two allusions you are talking about. Phoebus is a sun-god and Phaeton is the driver of the mythological chariot which contains the sun. These two mythic figures get mentioned because Juliet is talking about the sun, and they are connected with it. In very simple and coarse terms, this soliloquy means "Hurry up sun and set, so it will be night, Romeo will come and we can have some nooky." Naturally, Shakespeare puts much more beautiful and moving words than that in Juliet's mouth; it is one of the most beautiful expressions of carnal desire ever written.