I believe it's to do with snow compacting in the grooves.
Winter tyres in particular deal with snow by the soft compound compressing the snow in the treads, so that it can be released and thrown clear when the tyre rotates. If the tread depth is too shallow, this compression doesn't work and the snow actually sticks in the tread, significantly reducing traction.
Presumably, all season tyres are made from a harder compound, and rely on this action less, but I would imagine it would still have some impact.
Basically, bald tyres are not good in snow... I think we can work that bit out, but there is a technical reason why it's recommended that winter tyres should be changed at 4mm.