This is what I call the "representative spokesman convention", in which the spokesman repeats somebody else's words in the first person, as though they were his own.
The most obvious example is the simultaneous translator. The foreign President speaks to a press conference and says "I will declare war on everybody" in Russian (for example), and the translator sitting next to him immediately says "I will declare war on everybody" in English. Just "I will", not "He says he will". A less obvious example is the image of the apeaker on a television screen, which is nothing more than a collection of pixels, but still says "I will" as though it were the real person.
The angel of the Lord is another representative spokesman and tends to speak in the same way. Another example is the angel of the Lord speaking to Hagar and saying "I will greatly multiply your descendants" (Genesis ch16 v10). In fact Hagar regards herself as having seen God (v13), since the angel is about as much of the visible presence of God as the average person can stand. This is where it get closer to the "television image" analogy.
Incdentally, this is why John gets confused in Revelation ch22 vv8-9. The speaker is the angel of Jesus, another representative spokesman, not Jesus himself.