I'm asking this mostly because of the second episode of Band of Brothers in which it was portrayed. At the end of the episode there is a footnote saying that the Battle of Brécourt Manor is still taught to American troops as a textbook example of how to assault a fortified position.
If one looks at the casualties on both sides it's clearly an allied victory. But in watching the battle in the Band of Brothers episode and reading up on it nothing stands out about the battle for me. What did the American forces do so well during this battle?
you have to understand the difference in numbers, and how the military instructs its officers in the planning of operations.
The military will instruct its commissioned officers to only carry out attacks which are sure to win, and the best way of doing this is to gain a three to one advantage in manpower (in this situation). It's reckoned that the Germans had about 40-50 men in those trenches, which would mean that they really should have massed about 120-150 men to attack so as to be certain of victory.
So that fact that 12 paratroopers were able to overcome these considerable odds to win through good tactics, surprise, aggression in the attack and exemplary leadership is an edifying example. The battle is showcased not to demonstrate that the force ratios are unnecessary, but that good leadership and training is an exceptional force multiplier.
In all fairness, we should probably consider the disparity in training and mission between the two forces. Paratroopers were highly trained, aggressive volunteer infantrymen, who went into battle expecting to be outnumbered and to have to make up for materiel inferiority with elan. The German defenders were, in all likelihood, artillerymen, and not fallschirmjagers (there's a bit of confusion on this issue, with the initial after action reports stating they had engaged FJs, while I think it's been proven that there were no German paras in the area at the time the attack took place). For gunners, personal defense is not a priority, and I believe that is reflected in the program: initially, they're too busy serving their pieces to really react to what's going on, and by the time they do, the Americans are in their trenches with them.
To sum up, a small group of highly motivated, well-led American paratroopers took advantage of the general chaos of the situation to flank and assault an artillery battery lacking infantry support.
I will point out one more aspect of the battle: the Germans had imperfect information about the battle while the Allies had better intelligence on the situation. The American force knew roughly what they were going to be facing--you can see this in /u/CrazyMcCookie 's explanation of Winters' plan. However, the Germans in general had imperfect information as to what they were facing. With the American paratroopers scattered around Normandy along with fake paratroopers being dropped, there were reports of paratroopers everywhere but seldom was it confirmed that paratroopers were in a particular area. So, when their position got hit the Germans could not have known that it was an attack by a relative handful of men. Did the Americans have reinforcements? Were the Germans surrounded? Was this a raid or was it an invasion? Combine that with a hail of gunfire from a couple machine guns and you have a situation where a relative lack of information combined with a surprise assault can feel overwhelming--even if the attack comes from an outnumbered enemy.
During the Battle of Brecourt Manor American forces deployed a small number of soldiers against a larger, entrenched German force using textbook assault tactics. The tactic they used is commonly called Fire and Maneuver and involves one 'team' acting as a base of fire and providing suppression on an enemy position whilst a second 'team' assaults aforementioned position.
Elaboration and Explanation: Having scouted the enemy position in order to gain information on both the enemy force and it's orientation Richard Winters (1st Lt.) deployed his forces so that two machine-guns (M1919 30.cal) provided a base of fire suppressing enemy positions and allowing an assault team to close and engage the enemy from an advantageous position.
The combination of superior fire and a rapid and organised assault meant that the German forces were rapidly destroyed and American forces took minimal casualties.