# I will lay my vengeance upon Edom by the hand of my people Israel

"I will use my people Israel to punish the people of Edom". This refers to Israel's army. 

# they will do to Edom according to my anger and my fury

"they will show Edom my anger and fury" or "they will punish Edom according to my anger and fury against the people of Edom"

# my anger and my fury

"my furious anger" or "my extreme anger"

# this is the Lord Yahweh's declaration

"this is what I, the Lord Yahweh, have declared". Yahweh speaks of himself to express the certainty of what he is declaring. See Ezekiel 5:11.