Narrated Abu Sai’d Al-Khudri: “The Prophet (peace and blessings of Allah be to him) used to proceed to the musalla on the days of Eid al-Fitr and Eid al-Adha; the first thing to begin with was the prayer and after that he would stand facing the people and the people would keep sitting in their rows. Then he would preach to them, advise them and give them orders, (i.e. Khutbah or Eid sermon). And after that if he wished to send an army for an expedition, he would do so; or if he wanted to give an order, he would do so, and then depart. The people followed this tradition till I went out with Marwan, the Governor of Medina, for the prayer of Eid al-Adha or Eid al-Fitr.”
(Sahih Bukhari)
In Friday prayers we first listen to the khutbah (Friday sermon) and then perform the prayers in congregation. But in Eid prayers, the order is reversed: we first perform Eid prayers and then listen to the khutbah (Eid sermon). Khutbah is also part of the prayers; so it is incumbent upon the Muslims to listen to the khutbah.
Another hadith says that after delivering the Eid khutbah, the Blessed Messenger used to go to the women section and preach them. In the present circumstances, in large mosques where loud speakers are used, a speaker device can be installed in the women section so that women are equally benefitted with the sermon.
Allah’s Messenger never proceeded for the Eid al-Fitr prayer unless he had eaten odd number of dates.