The common cause of changing the wavelength of the light in this case is the rapid decay to the lowest vibrational level of singlet exited state. Furthermore, fluorescent dye generally decay to higher vibrational levels of singlet ground state. for more info you can search for the Stokes shift.
If you like this topic look abou:
ESIPT(Excited State Intramolecular Proton Transfer)
http://www.bmo.physik.uni-muenchen.de/~wwwriedle/projects/esipt_overview/esipt_overview.php
FRET (Förster resonance energy transfer)
Youtube demonstration
https://www.youtube.com/watch?v=3bmb_oDl6ws
If you want a trivial answer. Light of appropriate wavelength when it excites a fluorescent molecule, one gets fluorescence. So one sees the color of the beam changing when it is passing through the fluorescence solution because now we are actually seeing the fluorescence from dye molecules instead of the incident light. The fluorescence being of different color (lesser energy) we see it as different. This is not always true. If the incident light and fluorescence are of same color then we cannot make out the difference. For example, in gas phase the incident light and fluorescence of molecules are of same wavelength.
Also in rigid aromatic molecules the energy difference is less.
More technically it is because of Stokes shift.
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