How does glow-in-the-dark plastic "trap" light?

Here is a good article.

“All glow-in-the-dark products contain phosphors. A phosphor is a substance that radiates visible light after being energized.
“Phosphors have three characteristics:

The type of energy they require to be energized
The color of the visible light that they produce
The length of time that they glow after being energized (known as the persistence of the phosphor)

To make a glow-in-the-dark toy, what you want is a phosphor that is energized by normal light and that has a very long persistence. Two phosphors that have these properties are Zinc Sulfide and Strontium Aluminate. Strontium Aluminate is newer—it’s what you see in the “super” glow-in-the-dark toys. It has a much longer persistence than Zinc Sulfide does. The phosphor is mixed into a plastic and molded to make most glow-in-the-dark stuff.”

In short, the material absorbs the photon, and then releases it later.

Glow in the dark toys actually operate on a very simple principle, though the actual process is rather complicated. The general gist is as @Rarebear said, the photon is absorbed, and is released at a later point.

A little more granular, each of the substances @Seek_Kolinahr posted have the ability to absorb light. Most things do, actually. It’s the reason things heat up when exposed to the sun, as the photons are absorbed and converted to heat. Photons, in a general sense, are nothing more then packets of energy. And so, when absorbed by a substance, they simply cease to be. The energy is transfered to the other molecule.

Now, when glow in the dark objects are exposed to light, the same thing happens. The molecules absorb the light and the molecules enter what is known as an ‘excited state’. That is, it has more energy then it did before. This excited state’s duration depends on the material. Most materials like, say, cotton in a T-Shirt, leave this state very fast, and the energy is converted into heat. For Glowing objects, this state is much more stable, and so it can stay in this excited state for hours.

And that’s where the glowing comes in, along with the Phosphors @Seek_Kolinahr mentioned. These molecules are excited and have more energy. Though this is stable, they are not completely stable. If they were, it wouldn’t glow, as the energy would never leave. So, over the course of several hours (usually, sometimes less) after the glowing object is taken out of the light, these states lose stability. This is known as ‘decay’ of the excited state. It’s quite similar to decay of a radioactive material. In fact, they’re both known as ‘Radiative decay’. However, in the glowing case, instead of the decay causing harmful radiation, it releases the energy as another visible light photon.

So, the end result is that over a few hours, each molecule in the glowing object goes from this excited state back down to it’s normal state, and the difference in energy is ejected as a photon, and you have light!

Another way to look at what @BhacSsylan said is that all material, like the cotton T-shirt, radiate energy. It is just that glow in the dark material radiate in the visible spectrum, which you can see, whereas other materials radiate in the infrared spectrum, which you can not see.