In Genesis 16 the Angel of the Lord appears to Hagar, who later refers to this person as God Himself. Why does the Bible use the term "Angel of the Lord" instead of simply saying "God?"
The Second Person of the Godhead is called the Angel of the Lord in the Old Testament, and in fact all manifestations of God in the Old Testament are manifestations of the Angel of the Lord. These two are one and the same, and they are always appearances of the pre-incarnate Christ. The form of the appearance may change, but it's always the same Person (Christ) and it's always the Angel of the Lord. Therefore, this term is the Bible's name for Christ prior to His incarnation.
The Bible tells us that the Second Person of the Godhead is the member of the Trinity responsible for creating all things and He is also the only member of the Godhead Who enters into the Creation to become visible:
Col. 1:15 He is the image of the invisible God, the firstborn of all creation.
Col. 1:16 For by Him all things were created, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities — all things have been created through Him and for Him.
Col. 2:9 For in Him all the fullness of Deity dwells in bodily form,
Heb. 1:1 God, after He spoke long ago to the fathers in the prophets in many portions and in many ways,
Heb. 1:2 in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world.
Heb. 1:3 And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high,
Heb. 1:4 having become as much better than the angels, as He has inherited a more excellent name than they.
So the terminology may change from verse to verse, but the meaning is consistent. All appearances of God in the Old Testament are always the angel of the Lord, the pre-incarnate Christ, God Himself.
You might find the following article helpful: How do we know the angel of the Lord is Jesus?