Brining vs Salting - what's going on?
The question starts with an inaccurate assumption.  Salt does NOT necessarily result in "drier steak."  Salting a steak will draw some moisture out of the outermost layer of meat initially.  That's true.  However, salt also helps to break down the outer muscle structure of the meat.  Thus, within 10-15 minutes after meat is salted, water will actually start moving back into the meat, and after somewhere around 45 minutes to an hour, all of that "lost moisture" will be reabsorbed.  (I covered all this in more detail, with some links, in an answer to a previous question.)

If you use a rub and allow the salt to penetrate at least an hour or so, you won't lose moisture and will actually end up with a more flavorful outer layer of meat.

And yes, salt used long-term, like in dry-curing, will eventually dry out meat more.  Salt attracts water, so it will gradually draw water out of meat, but then the water on the surface will evaporate.  Thus, the salt will draw out even more moisture to the surface, eventually drying and curing the meat.  But this is a process that happens over days, weeks, and months, as water migrates much more slowly internally within meat.  

As for brining, the reason why it increases moisture is because physical and chemical systems like to come to equilibrium.  Higher salt content outside the meat means that the salt will gradually flow ("diffuse") inside  the meat to equalize the concentration.  But salt is attracted to the water molecules it is dissolved in (the same reason it draws out moisture in dry curing), so water comes along with it, thus carrying in moisture.  In the case of curing, there's no water for the salt to carry in with it, so you just end up eventually with a dry, salted hunk of preserved meat.  But in brining, there's excess water, so it's attracted and follows the salt inward.  The process is largely diffusion, but osmosis plays a role too.  (And I've also said way more about this in another answer.)