I don't know the answer to this question for sure, but I can offer a suggestion. Ultraviolet photons contain more energy than visible photons. The sensitive material in the beads is likely to be composed of atoms or molecules with certain electrons not able to absorb photons unless the photons have an energy at least as great as that corresponding to the ultraviolet. Once absorbed, either the energy is released by the electron emitting light of less energy (a visible color) than that absorbed (this is called fluorescence), or the structure of the material is changed by the absorbed photon so the "activated" material selectively absorbs or
scatters visible light of different wavelengths, resulting in the observed coloration. If you contact the company that makes the material, and they give an explanation, could you let us know what it is?

I first encountered UV beads while taking a biology course in Antarctica. One of the instructors passed out zipper pulls for everyone which had UV beads braided into them, so that we could tell when our chance of sunburn was greatest. As you know, the ozone hole is worst over the Antarctic continent, so UV exposure is higher there than anywhere else on the planet (at sea level, anyway). Dr. Karentz studies the damage caused by this increased UV exposure to Antarctic marine algae, and the natural sunscreen these algae make (called microsporine-like amino acids, or MAA's). I forwarded your question
on to her in Antarctica. Here is what she had to say:

The course this year is going great, just like last year. We are finishing the theme rotations and everyone is moving into project mode. As a coincidence we were just talking about the UV beads on the zipper pulls this morning. I tried a few sites on the Internet, but there does not seem to be much information available (although a search for "UV beads" does turn up over 600 sites). I do not know what the compounds are, but each color must be a different compound with a chromatophore that absorbs a different range of UVA wavelengths.

1) Visible light is a form of electromagnetic radiation.
2) Electromagnetic radiation is the type of radiation that is produced when electric charges change the speed at which they are moving (are accelerated or decelerated), or when electrons jump from high to low energy levels within atoms or molecules.
3) The main characteristic of electromagnetic radiation is the energy it carries. Depending on that energy, the radiation interacts with matter in different ways. Scientist classify the radiation also according to its energy.
4) The human eye is sensitive only to a very small interval of energies of electromagnetic radiation. That interval is what we call visible light.
5) Electromagnetic radiation with energy immediately higher that the high end of human perception is called ultraviolet radiation. Radiation carrying energy immediately lower than what we can see is called infrared radiation.
6) Certain dyes are capable of absorbing UV radiation. This absorption promotes electrons to higher energy levels within the molecule, creating that is called an excited state. Molecules generally stay in an excited state for very short periods of time. On returning to the unexcited or ground state, the molecule emits (gives back) energy. The energy is emitted as electromagnetic radiation. The energy of that radiation will be the difference in energy between the excited and the lower states. For some molecules, there are several lower states between the excited state and the ground state. If it happens that the difference between the excited state and one of those lower states corresponds to the energy of visible light, we will have the bead glowing under the influence of uv.

I don't really know exactly how these work. I assume the UV radiation ionizes something in the plastic and thereby changes the optical properties.