Sheikh Attiya Saqr- May Allah have mercy on him- says: It is desirable that one eats before he leaves home for Eid al Fitr prayer, however this is not the same for Eid Al Adha.
Following the example of the Prophet ﷺ as narrated by Ahmad and others, it is upon a person to follow the footsteps of the prophet ﷺ and eat 3 or 5 dates, before he goes out for Eid ul Fitr prayer. That is because the day before he was fasting, and he did not have breakfast. On the night of Eid and on this day, he is hosted by the most Merciful, and it is forbidden to fast on this day. It is preferred to break the fast as soon as possible. As for Eid Al Adha, it is preferred to break the fast later, until he has prayed the Eid al Adha prayer and has slaughtered his sacrifice and eats from it if the sacrifice meat if he has one.
As for this ruling on the prophet ﷺ going to Eid prayer using one route and would return from the prayer from another, Bukhari has narrated from Jaber (RA) that he said: On the day of Eid the Prophet ﷺ used to return through a way different from that by which he went.
Muslim and others have narrated from Abu Hurairah (RA) that he said: If the Prophet ﷺ went for Eid using one path, he would return using a different path from the one he used on the way there.
The wisdom behind this as the scholars have said is that the two roads will be witnesses for him. It is known that on the day of judgement, the earth will be a witness for us, or against us as Allah SWA says : ( And the earth brings out its loads. And the human being says, “What is the matter with it?” On that Day, it will convey its chronicles. Because your Lord will have inspired it.)
For a good deed a person needs a witness, a human or other, like the caller to prayer has all those who hear his call become a witness to it, whether they are human, jinn, stone, tree, or earth.
Therefore, the wisdom behind using different roads on the day of EId is so that the road can bear witness for the person, or the two roads may bear witness for him and that is better for him on the day of judgement.