# I will break the Assyrian in my land

"I will defeat the Assyrian in my land" or "I will cause the Assyrian in my land to be defeated" 

# the Assyrian

"the king of Assyria and his army" or "the Assyrian army"

# trample him underfoot

This represents completely defeating him. 

# Then his yoke will be lifted from off them and his burden from off their shoulder

"Then I will lift his yoke from them and his burden from their shoulder" 

# his yoke ... his burden

The word "his" refers to Assyria.

# from off them ... from off their shoulder

The words "them" and "their" refer to the people of Israel.