As Vladimir said, it's not cold weather per se that makes static buildup easier, it's dry weather. It just so happens that, for a variety of physical chemistry reasons, cold air can't have as much water in it as warm air, and so it is naturally drier in cold temperatures than in warm.
As far as what that does, it prevents a slow, gradual discharge of the electrons your body builds up by friction. So, it doesn't cause static buildup, but it facilitates it.
What actually causes the static buildup is the physical interaction between differing materials. A handy quick reference is the Triboelectric Series, showing common materials that will readily transfer electrons. Anodic Index is another common tool for metals, showing their relative propensity to give up electrons permanently under an electrical voltage, and galvanically corrode.
Basically, human skin is the most "electropositive" everyday material, because of the electrolytes in your cells such as sodium, calcium and magnesium. These elements readily give up electrons to form a positive voltage in your skin. That voltage can get quite high; the average static spark is in the low tens of thousands of volts. The reason the average static spark isn't really damaging (painful, yes, but you get over it in a couple seconds instead of needing to go to the hospital) is because of the relatively high resistance of your skin (which also helps to hold that charge; your skin becomes a "dielectric", like the materials in a capacitor), which can be in the high hundreds of thousands of ohms, meaning the actual amount of current moving during a static spark is normally in the milliamps.
Other materials either give up electrons less readily, or accept them more readily. This transfer of electrons can be accomplished with ordinary physical interaction, like rubbing the two against each other. Many plastics, like PET (polyester), polystyrene, PVC, Teflon, etc, will readily accept electrons.
When you rub a material that readily donates electrons against a material that readily accepts them, the energy from the friction is enough to cause the electron transfer. The materials, with opposing charge, will then attract, and if the charge is high enough, will arc.