The theory is that crowd noise hurts the offense and helps the defense, which is most critical on 3rd down, where the offense is usually facing their last chance for a 1st down.
The offense has to coordinate several things in order to make a successful play. The play caller (usually the head coach or offensive coordinator) needs to select a play from the sidelines and communicate that play to the quarterback, usually via a headset radio. The quarterback then has to communicate that play to the rest of the offense, usually verbally in a huddle. Finally, the quarterback may need to change the play at the line, which he communicates by yelling an audible call to the team.
Loud crowd noise can disrupt any of these communications.
Teams like Seattle and Minnesota have a reputation for having very loud stadiums. Seattle believes in the advantage that their crowd gives them so much that they refer to the fans as the "12th Man" and have retired jersey number 12 in honor of the fans. Minnesota and other stadiums have been accused of pumping amplified crowd noise through stadium loudspeakers to gain an advantage.
In 1989, the NFL felt so strongly about the disadvantage that the visiting offense faced that they instituted a penalty for excessive crowd noise, issued against the home defense. The referees in enforcing the rule could take away timeouts and issue penalties, 5 yards at a time, until the crowd quieted down. The new rule also banned stadiums from encouraging noise via announcements. The rule was not very popular, and officials eventually stopped enforcing the rule. The penalty was officially dropped from the rulebook in 2007, and video scoreboard announcements encouraging noise were allowed again in 2013.
A sound effect played by the stadium before a third down is simply a cue to the crowd that the visiting team is in a third down situation and that they should be loud to help their team. Different stadiums have different traditions for these sound effects: Denver has a bronco neighing, Minnesota has a Viking horn, Jacksonville has a jaguar roaring. They usually play these after a good play by the home team, but they can also be used to alert the crowd to a third down situation.