I'm really excited for this because I get to reference the almighty color wheel!! Fair warning, this answer is much more qualitative than quantitative, but that's more interesting sometimes anyways.
If you look at the structure of the chromate and dichromate ions next to each other (see here for structures: https://en.wikipedia.org/wiki/Chromate_and_dichromate), the only major difference between the two is that the Cr-O bond joining the two chromate ions (missing an oxygen) is now a single bond. This means that bond will vibrate at a lower frequency, and because frequency and wavelength are inversely related, that bond will absorb a longer wavelength of light.
Now to the color wheel! It is a general chemistry (often unexplained) fact that the color we see is the complementary color of the wavelength of a bond's vibration. Thus, in the case of the chromate ion, we see yellow, and across from yellow is the purple-ish region. That means, if one of the bonds in the chromate ion, and thus two of the bonds in the dichromate ion, were absorbing a longer wavelength like we said earlier, on average we would expect something just longer than purple-ish, like blue, to be absorbed.
The complementary color of blue is red slash orange, and that is in fact the color we see in the dichromate ion!
At the heart of all this is the principle that the colors we see are those wavelengths of light which on average are not absorbed by a large number (on the order of Avogadro's number) of molecules.
An approach like this will only be reliable for very similar molecules like the two we have here.