בשר basar [H:1320] is the Hebrew word for "flesh," the skin or meat of animals or man; and when used in the phrase "all flesh" it means "all mankind." The verbal root of this word can be found in the following passage where it is translated as "proclaim the good news."
The spirit of Adonai Yahweh is upon me because Yahweh has anointed me to proclaim the good news (basar)... Isaiah 61:1
What does "flesh" have to do with "proclaiming good news"? Simple-when one proclaimed good news, such as the arrival of a new baby or visitors, or some other celebration, an animal was slaughtered and "flesh" was served.