The dye they used is called SYBR Green I. It's a molecule that binds to DNA and only fluoresces at a specific wavelength, but it only fluoresces once it's bound to DNA and exposed to a certain wavelength of light. You can observe the presence of bound molecules using spectroscopy which indicates whether or not DNA is present.

There are other fluorescent dyes, eg BigDye, which are used for genetic sequencing. You won't see anything with the naked eye, but an automatic sequencer can detect them.