Why does a brine wash help reduce water from the organic phase?
An extraction is a separation of material based on it's relative solubility in two different solvents. Normally, these two solvents are idealized to water and octanol and it is given that since these are immiscible solvents, there is no water in the octanol and no octanol in the water.
Once you start adding other components, or using more polar organic solvents such as chloroform or ethyl acetate, you do get some forces attracting water into the organic layer, and as you'e noticed if you've tried to strip one down, it can be quite a bit of water. One answer to this is to use a solvent that is even more attractive to water than plain water, and a good cheap one is a saturated sodium chloride solution. As you can tell from osmotic pressure experiments, strongly ionic solutions will pull water molecules away from pure water. It turns out they also pull water away from organic solvents that have water entrapped either due to some limited solubility of water in the solvent or to hydration of various solutes that might be present.
Adding salt to your water also does some other very useful things in extraction, like limiting the solubility of organic materials in the aqueous layer and breaking emulsions, but for that final wash with sat'd sodium chloride to remove water before using a drying agent, this is the effect in play.