My understanding is that the main active ingredient in zicam is a form of zinc and it's simply a homeopathic product. This is purportedly expected to block adhesion of the rhinovirus to local nasal receptors, prevent capsule formation around the virus (capsules make it harder to find/destroy the bug in question), as well as possibly potentiate the abilities of some antibodies in your immune system. I've looked at some studies and they are all pretty variable. None seem to suggest a lessening of degree of symptoms, but one suggested that there was a statistically significant reduction in the duration of the cold (by as much as a third).
There can be some side effects like local anaphylaxis, headache, etc.
Overall, if there is any value to the therapy it will likely be highly individualized. It will definitely have to be taken within 24 hours of the beginning of symptoms, however, for it to be effective at all (the earlier, the better). I don't really have any first hand information on that topic.
