We don't execute the Zaken Mamre (rebellious elder) in the Beit Din (court) in his own town. If he returned to his own town and told people Halachic (pertaining to Jewish law) rulings that were contrary to the Beit Din, we do not judge him in that court because the Pasuk (verse) tells us that he specifically has to be judged and put to death in the place that Hashem chose, which is Yerushalayim. And also, you're not allowed to execute him in Yavne. 40 years before the destruction of the second Beit HaMikdash, the high court, the Sanhedrin, left the spot of the Beit HaMikdash and went to different locations, one of them being Yavne.
So the situation of the Mishnah would have to be that he was told by the Beit Din of Yerushalayim not to give a certain Halachic ruling. Then he returned home and didn't defy their ruling for many years. By the time he violated it, the Beit Din was no longer in Yerushalayim, but in Yavne. So what we see from here is that even though he was told by the Beit Din of Yerushalayim not to do something, if the Beit Din was no longer in Yerushalayim but back in Yavne, the Beit Din cannot try him and put him to death, because it's also about where the execution is going to happen.
Ela ma'alin oto (Rather, they bring him up) to the Beit Din HaGadol (High Court) of Yerushalayim: he has to be in Yerushalayim in order for the execution of the Zaken Mamre to happen. He has to be brought to the High Court, the Sanhedrin Yerushalayim, and they guard him there until the next holiday of Pesach, Shavuot, or Sukkot, when the entire nation is going to be in Yerushalayim. They execute him during the regel, during the holiday, like the Pasuk tells us. Like the Pasuk tells us, the entire nation is going to hear about it, and they're going to be afraid, and they're not going to act in this way against the Beit Din anymore.
This is the opinion of Rabbi Akiva. Generally, the Beit Din has to put somebody to death right away, not cause them agony by prolonging the process of sitting there waiting to be put to death. The Beit Din would usually end their life right away. But according to Rabbi Akiva, in this unique scenario, we need the execution to happen in order to instill fear in everybody else that they can't defy the ruling of the Beit Din. So it's an exception to the rule, and they would wait until everyone else is around on the regel.
Rabbi Yehuda says it's not proper to delay the judgment until the regel. It can be months until the next holiday arrives, and we don't want this person lingering and causing him unneeded suffering. They execute him right away, says Rabbi Yehuda. They write messages and send messengers all over the country, wherever Jews live, in order to tell them what happened. And they tell everyone, “So-and-so, the son of so-and-so, was sentenced to death by the Beit Din because he was a Zaken Mamre. He violated the ruling of the Beit Din.” According to Rabbi Yehuda, that fulfills the obligation of everyone having fear and being nervous about violating the ruling of the Beit Din. If they hear about it, they'll hear that an execution took place. They're not going to violate the ruling of the Beit Din; they'll learn from this case, and there's no need to wait for the regel.