The material doesn't emmit the light in the dark, it just does it slowly over a long period (you can probably only see it in the dark. So for example it will absorb a photon, and then every minute there is a 50% chance it will emit the energy again.
The way it does it is to do with energy levels, in an atom or molecule the electrons are only able to have certain energies. In phosporescence you have 3 energy levels, an electron absorbs a high energy (blueish) photon which pushes it up to a high energy level. Some of the electrons fall straight back down again, but others loose a bit less energy and end up into an intermediate energy level from which it is very hard to fall back down to the original low energy ground state. You could think of it like for the electron to loose energy the atom the electron is in, has to be wobbled by something else in exactly the right way. This means that the light is given out slowly and if you take the object into a dark room you can see it glowing.
-----------------------
^ | Easy transition
| V
| --------------
| ! /
| ! / photon given off
| ! /
| ! Unlikely transition
| V
-------------------------- Low energy Ground state