This is necessarily a speculative answer, and I think it cannot be otherwise.

1. It's the deadliest extermination camp, with about 1 million victims. Treblinka is the second deadliest, with about 800,000 victims. Earlier the death toll was often thought to be even much higher.
2. Unlike Treblinka (and most other extermination camps) the Auschwitz complex wasn't fully demolished by the Nazis. You can actually see many original or quasi-original (somewhat modified) structures, like the crematorium in the main camp, the ruins of the denotated Kremas in Birkenau, the barracks, the infamous main gate at Birkenau, etc. So it's the most visual surviving extermination complex. There's also Majdanek, where you can see quite a lot, but its death toll is much lower.
3. Due to the many camp structures surviving, they were filmed by the Soviets; the liberation footage, also showing some survivors, was used in innumerable documentaries.
4. Many tens of thousands actually went through Auschwitz and survived (lots of them non-Jews) one way or another (e.g. tens of thousands of Jews were transited through it to slave labor camps, some worked in Auschwitz, some were Mengele's twins etc.). Not so with most other extermination camps, so many more testimonies are available for Auschwitz, giving it a special prominence (not to mention that numerous Auschwitz survivors went on to be quite notable for their own achievements).
5. Auschwitz figured prominently during the International Military Tribunal at Nuremberg due to the commandant Rudolf Höß' testimony. Other extermination camps were testified about too, but comparatively shortly.
6. The image of the industrial extermination, with the crematoria with their smoking chimneys as the factories of death, must have been particularly effective at the time. Most other extermination camps did not have crematoria.

Others have answered this question, but I also addressed it to some degree here.

Your question asks about Auschwitz-Birkenau among Extermination Camps, but if we open that up to Auschwiitz among other camps, I will add a bit.

In addition to what others have said, however, it should be noted that Auschwitz was not always the most discussed camp. From liberation to a number of years after the war, Dachau would have been more recognizable to Americans (Bergen-Belsen to British). This is because each of those camps was liberated by the respective armies. Since Auschwitz was liberated by the Soviets, less was known about it among Americans.

I suspect, though haven't found any direct evidence to prove it, that many of the confusions of the years immediately following liberation were caused by the mass transfers of prisoners from Auschwitz to numerous camps inside Germany. As Western armies liberated those camps and heard stories of Auschwitz, where many more prisoners had been kept, they associated them with the camp at which they liberated those prisoners. In other words, I believe it likely that when prisoner x, who had been transferred from Auschwitz but was liberated at Dachau, spoke of crematoria and gas chambers, the Allied troops thought that prisoner x meant gas chambers at Dachau. Hence why some believed that such chambers were used at Dachau to kill prisoners. Again, I believe this to be the likely origin of the commonly held belief of the 1950s that Dachau had used gas chambers to kill people.

To your point on extermination camps specifically, I simply reiterate what others stated. The other extermination camp were 1) torn down and destroyed by the Nazis rather than liberated by the Allies and 2) Auschwitz (as a camp with multiple rather than a single function) had survivors where the others had so few.