The underlying reason that the sky appears ultraviolet to a bee (and blue to us) is that something happens to the sun's rays when they hit charged particles. These charged particles are in the ionosphere and in Nitrogen and Oxygen "dipoles" in the atmosphere. The sun's radiation will cause the charged particle to vibrate, and when the charged particle vibrates it re-radiates energy. If you work out the mathematics (physics) of this re-radiation, it turns out that the energy is re-radiated in proportion to the fourth power of the frequency (as discovered by Raleigh in the 1800's and worked out with a more accurate distribution formula by Einstein early in the 20th century). For humans, who have eyes that have blue, yellow and red sensors, the fourth power formula means that blue from the sky is ten times stronger than red, so the sky looks blue. Similarly, for bees (and other insects), the highest frequency sensors in their eyes detect ultraviolet), and so the sky appears ultraviolet to them.
Sources: Karl von Frisch on Bees (e.g. in Answers.com); basic courses on electrodynamics and magnetism (or Google "Why is the sky blue").