Why does soda water make plants grow faster?

Whether a plant will grow faster if given soda water rather than tap water depends on the plant's needs and the content of the water. Certain additives in the soda water may be beneficial for some plants, but not all. Note also that flavored and colored sodas have different ingredients than plain soda water.
Plain soda water tends to be alkaline, therefore plants that prefer an acidic soil may be harmed by soda water. Alkali-loving plants include cucumbers, clematis and lilacs. Talk to an experienced gardener or other expert before experimenting.

Regular soda water (as opposed to sweet drinks called sodas in some countries) contains only water and carbon dioxide. It is possible that the carbon dioxide is released from the soil producing a carbon dioxide rich atmosphere locally. Plants absorb carbon dioxide to build tissue using photosynthesis. Note that soda does not mean salt as stated in the first answer, this is quite wrong. Sometimes sodium carbonate is called soda, or washing soda. This is not a component of soda water and has no known beneficial effects on plants.