Why does Brining help food to retain water, but adding salt will draw the moisture out?

For the brine, it's because of osmosis When you have a semi-permeable membrane, like a cucumber skin, water will tend to move from the higherlower solute mixture (the salt water) to the lower higher solute mixture (the water with organic material inside the cucumber). This will cause the cucumber to absorb water AND some salt, until the point where the water in the cucumber is as soluble as the surrounding brine.

When you add solid salt to an item, steak for example, Osmosis is no longer at work. Instead, you're dealing with absorption , a completely different chemical process.

Actually, osmosis would work exactly the opposite way of how Mike Sherov suggests: it would draw water out of the cucumber (or the turkey) and into the brine. (Close reading of the Wikipedia article bears this out.)

(Sorry for posting this as an answer - apparently I can't reply to Mike's answer directly, yet. I don't know the actual answer.)

Your assertion that "when you cook with salt it causes the foods to get less moist" is a myth. Both brining and salting increase the moistness and flavor of meat.

"...even if there is no surrounding water to draw in, proteins are modified by the {salt} ions in ways that cause them to bind the water in the flesh more tightly -- as well as to resist the shrinking of muscle fibers that squeezes juices out during cooking. The flesh continues to swell and bind water more tightly until its salinity increases to 6%, and then it shrinks and begins to lose water." Modernist Cuisine, Volume 3, Page 154

Numerous recipes and articles from America's Test Kitchen have also demonstrated this principle as well. "Dry salting" may take more time to accomplish the same feat because water must first diffuse out of the meat to dissolve the salt before it can begin the absorption process.

Salting above a 6% salinity will draw out moisture, but meat is unpalatably salty around 1% salt by weight.

A thorough explanation with slideshows is available at:

https://stellaculinary.com/podcasts/video/the-science-behind-brining-resource-page

what mike says is correct. This process can be used to make meat more tender, if you put meat in saltwater for some hours it will absorb about 10% of its weight in water. So if you put a 100gr piece of meat in saltwater it will absorb about 10g of water. It will leave the meat salty, but more tender. Removing salt from the other components of the dish will hopefully balance things out.