What was the purpose of the Battle of Fredricksburg?

Answer 1

Gen. Burnside had replaced McClellan due to McClellan's constant slowness - even though McClellan defeated Lee at Antietam, he still didn't press any advantages. Burnside, understanding that he had been promoted to go out and move quickly, chose to try a feint, followed by a quick strike to Richmond, hoping he could confuse Lee long enough to get the jump on Lee.

Lee, on the other hand, was still recovering from Antietam, and simply was preserving Richmond from Union attack.