The pathogenesis of gout is difficult and controversial, but it fundamentally involves an inflammatory response to urate crystals that are deposited in and around the synovium, tendons and articular cartilage.
Colchicine has the useful effect, in this regard, of inhibiting the migration of phagocytes to these areas during an acute attack. Alone, this would obviously reduce local inflammation, but stopping phagocytes doing their thing also has further indirect benefits. In places of inflammation, the phagocytes power themselves via the rapid oxidation of glucose. Some of this is unavoidably turned into lactate (oxygen can't be provided quickly enough to prevent anaerobic metabolism) and this in turn lowers the pH. In other cases this might be pretty much incidental, but in gouty joints this increased acidity is harmful: it causes even more urate cystal precipitation in the joint ... and thus more inflammation too.
Colchicine, by preventing phagocyte migration, therefore prevents the inflammatory response of acute gout via both a direct and an indirect method.
Apart from in gout, colchicine can also theoretically be used as a chemotherapeutic agent. It binds to tubulin, a major constituent of microtubules. Since microtubules provide all the directions for the cell's parts during mitosis (among other things), colchicine thus prevents mitosis. As with other non-specific chemotherapeutic agents, this simply has the effect of blighting the fates of the most rapidly dividing cells. In most cases these cells are cancerous, but other rapidly multiplying normal cells will suffer too, and this accounts for many of colchicine's side effects.