Am I right in assuming they're used to bring the range of output from the photodiode up to the range of inputs for the analogue pins?
Yes
If I tried first just directly connecting each anode to separate analogue pins, powering each anode separately from 5v with a resistor in series and the common cathode direct to ground, would I be in danger of damaging anything?
No
making the whole project much simpler.
Yes but the range of readings you are likely to get is small which might hamper the resolution of the colour recognition.