"Now the time had come for Elizabeth to give birth, and she gave birth to a son. Her neighbors and her relatives heard that the Lord had displayed His great mercy toward her; and they were rejoicing with her."
Elizabeth gave birth to the son God had promised. When Elizabeth's neighbors and relatives heard what God had done, they rejoiced with her. God always keeps His promises. It's to His own glory to do so. "[God's] glory lies in His goodness, His moral character, His justice, benevolence, holiness, mercy." -- Charles G. Finney. When we rejoice with other Christians over what God has given them, we have a chance to do more than celebrate the blessing they have received. We can bring glory to God, the Giver of the blessing. We do this by giving Him honor, by praising and worshiping Him. God’s glory is the essence of His nature, and we give glory to Him by recognizing that essence.