The colors produced by transition metal ions in solution are due to the "splitting" of the degenerate energy levels of the d-orbitals in the atom/ion.
In isolation, the d-orbitals all have the same energy; however, if the atom/ion is surrounded by charged ions or partially charged ligands, the charges in the coordinating ions/ligands produce an electric field around the metal ion that is not spherically symmetric. This electric field causes some of the d-orbitals to have higher energy than in the isolated state, and others to have lower energy. This is called "ligand-field splitting (or, when this same effect occurs for transition metals in crystal, "crystal-field splitting". The same thing happens with f-electrons in the lanthanides (rare-earth elements) and actinides.
The difference in energy between the "split" orbitals often corresponds to the energy of a visible-light photon. Absorption of specific wavelength (energy) photons from white light will make that light no longer white (it will be colored).
Absorption of a photon with the same energy as the difference between the split energy levels of the valence electrons in a complexed ion causes an electron in the lower energy level to be "excited" to a vacant orbital of higher energy.
Zn2+ is a d10 ion, meaning that there are 10 d-orbital electrons. There are 5 spatial d-orbitals, each capable of holding 2 electrons (1 spin-up and the other spin-down), for a total of 10 electrons. The d-orbitals in a Zn2+ ion are therefore completely full. Even though the orbitals in a complexed Zn2+ ion are split in energy, and may have an energy difference corresponding to the energy of a visible-light photon, an electron exited from the lower energy orbitals has no place to "land" in the higher energy valence orbitals (there are no vacancies in the higher energy orbitals), so this photon-absorbing process cannot occur. The next available excitation would be to some higher energy orbital, which would absorb a higher energy (shorter wavelength) photon, which presumably is outside the visible spectrum.