Well, codebreaking in general in WWII was accomplished in large part due to Alan Turing and the codebreaking of the Enigma code. However, I'm not a codebreaking expert so someone else should answer that part of the question ;)

However. With regards to the Unternehmen Wacht am Rhein, known stateside as the Battle of the Bulge (due to the salient created by the advancing German armor), the Allies actually didn't gain valuable strategic or tactical information from codebreaking nor any other espionage prior to the offensive.

While the Unternehmen Wacht am Rhein was doomed to fail, mostly for logistical reasons (the German offensive in the Caucus region of Soviet Russia had failed, including the complete destruction of the 6th army, depriving the Wehrmacht of VALUABLE oil), the SS Panzer divisions had enormous initial success.

This was largely due to the ignorance of Montgomery, but it wasn't really his fault. The Ardennes is a heavily forested region, and the Germans pulled off a feat of combat engineering to move their army through it. With regards to intelligence, the Americans and British were largely caught off guard due to excellent German countermeasures. Remember, at this time they had retreated back to Germany, and were defending with vigor (Operation Market Garden had recently been a colossal disaster for the Allies). The French Resistance during the occupation could get intelligence, cut phone lines, pose as Germans, etc, to gain knowledge of troop movements and tactical and strategic planning. But now, the Germans were defending their home soil, and there weren't many defectors. Finally, they exclusively used telephone communication rather than radio communication, because they had telephone lines in their country that were yet undamaged.

This perfect combination of a good choice by the German high command and the fact that they were on home turf allowed the Battle of the Bulge to be a huge initial success for the Third Reich.

Hope a codebreaking expert chimes in! Would love to hear more about that!