# He has raised up a horn of salvation for us in the house of his servant David

An animal's horn is a symbol of the animal's power to defend itself. The Bible often calls a powerful person a horn. Here, to raise up someone means to bring him into existence or to enable him to act. Alternate translation: "He has brought to us someone in the house of his servant David who has power to save us" 

# in the house of his servant David

David's "house" here represents his family, specifically, his descendants. Alternate translation: "in the family of his servant David" or "who is a descendant of his servant David"