How does colchicine prevent mitosis exactly?

No, it prevents the formation of the spindle in the first place. It binds to tubulin, which prevents it from polymerizing into microtubules. Without microtubules, the chromosomes line up along the centerline of the cell, but don't get pulled apart to the poles. Since the chromosomes can't be pulled apart, it never passes the metaphase checkpoint, and halts in its cycle. Immediately after adding colchicine, you won't see a change in the cycles of other cells, but over time, they'll all get stuck in metaphase. Once you remove the colchicine, the cells will pass into anaphase - they'll all proceed at once, and their cell cycles will by synchronized for a little while.