How does sodium in one's diet affect blood pressure?
The basic reason is osmosis, the tendency of solutes to move from an area of high concentration to one of low concentration across a permeable barrier. So, ingesting large amounts of salt results in a high sodium concentration in the blood stream. This in turn causes water to enter the blood vessels by osmosis. More water in the blood means a greater volume of liquid and, therefore, higher blood pressure. 

This process occurs primarily in the kidneys, specifically in the part of the nephron called the loop of henle (the image below was taken from wikipedia and is originally from Grey's Anatomy):



With low salt concentrations, water exits the loop and is retained by the body. When sodium concentration is high, the opposite occurs, water enters the blood stream by osmosis and increases blood pressure.