Full disclosure: I’m writing this because I painted my nails an amazing fluorescent orange, and it made me curious…
TL;DR: It’s UV rays!
Now for the science.
When light hits an object, the object absorbs some of the light and reflects the rest. Our eyes pick up what is reflected through cone cells in our retina. Objects appear to be different colours due to the chemical, atomic and material make-up of the object, alongside the external environment; all of this determines which wavelengths of light are absorbed and which are not.
For more on how the environment can affect colour, check out this video of how depth of water affects colours. Water absorbs some of the light before it hits your eyes, making colours appear paler and less vibrant.
When you look at a fluorescent object, the light reflected into your eyes includes UV light. This is normally just beyond visible violet in the colour spectrum, but certain dyes contain molecules which essentially convert the UV light into visible light.
This fluorescence is caused by electrons, which make several ‘downward transitions’ after an ‘upward transition’. When light hits the electron, it absorbs the light, transitioning it upwards into a high-energy state. It then makes several downward transitions to a lower-energy state, and releases this excess energy as visible light. The more visible lightwaves which enter your eye, the brighter the colour appears.