I take it as being they thought Peter's guardian angel had assumed his appearance to talk to them. We should note that this interpretation does not mean the Bible is teaching that guardian angels can assume their wards appearance but that these believing Jews in the first century believed such to be true.
The understanding of Peter's aggelos being his guardian angel with Peter's likeness did not appear in the English translations without precedent. Apparently some ancients believed that angels, especially a person’s guardian angel (Matthew 18:10), could take on the characteristics of a person.
The apocryphal writing Tobit 5:4-13 contains one such example as Tobias speaks with Raphael, not knowing that Raphael is an angel until later in the conversation.
4: Therefore when he went to seek a man, he found Raphael that was an angel.
5: But he knew not; and he said unto him, Canst thou go with me to Rages? and knowest thou those places well?
6: To whom the angel said, I will go with thee, and I know the way well: for I have lodged with our brother Gabael.
7: Then Tobias said unto him, Tarry for me, till I tell my father.
8: Then he said unto him, Go and tarry not. So he went in and said to his father, Behold, I have found one which will go with me. Then he said, Call him unto me, that I may know of what tribe he is, and whether he be a trusty man to go with thee.
9: So he called him, and he came in, and they saluted one another.
10: Then Tobit said unto him, Brother, shew me of what tribe and family thou art.
11: To whom he said, Dost thou seek for a tribe or family, or an hired man to go with thy son? Then Tobit said unto him, I would know, brother, thy kindred and name.
12: Then he said, I am Azarias, the son of Ananias the great, and of thy brethren.
13: Then Tobit said, Thou art welcome, brother; be not now angry with me, because I have enquired to know thy tribe and thy family; for thou art my brother, of an honest and good stock: for I know Ananias and Jonathas, sons of that great Samaias, as we went together to Jerusalem to worship, and offered the firstborn, and the tenths of the fruits; and they were not seduced with the error of our brethren: my brother, thou art of a good stock...
21: For the good angel will keep him company, and his journey shall be prosperous, and he shall return safe.
The Shepherd of Hermas (an early Christian book that was not recognized as canonical but was still deemed useful for reading) also has an angel appearing in human form.
Shepherd of Hermas Vision 5:1-7. 1 As I prayed in the house, and sat on the couch, there entered a man glorious in his visage, in the garb of a shepherd, with a white skin wrapped about him, and with a wallet on his shoulders and a staff in his hand. And he saluted me, and I saluted him in return.
2 And he immediately sat down by my side, and he saith unto me, "I was sent by the most holy angel, that I might dwell with thee the remaining days of thy life."
3 I thought he came to tempt me, and I say unto him, "Why, who art thou? For I know," say I, "unto whom I was delivered." He saith to me, "Dost thou not recognize me?" "No," I say. "I," saith he, "am the shepherd, unto whom thou wast delivered."
4 While he was still speaking, his form was changed, and I recognized him as being the same, to whom I was delivered; and straightway I was confounded, and fear seized me, and I was altogether overwhelmed with distress that I had answered him so wickedly and senselessly.
5 But he answered and said unto me, "Be not confounded, but strengthen thyself in my commandments which I am about to command thee. For I was sent," saith he, "that I might show thee again all the things which thou didst see before, merely the heads which are convenient for you. First of all, write down my commandments and my parables; and the other matters thou shalt write down as I shall show them to thee. The reason why," saith he, "I command thee to write down first the commandments and parables is, that thou mayest read them off-hand, and mayest be able to keep them."
6 So I wrote down the commandments and parables, as he commanded me.
7 If then, when ye hear them, ye keep them and walk in them, and do them with a pure heart, ye shall receive from the Lord all things that He promised you; but if, when ye hear them, ye do not repent, but still add to your sins, ye shall receive from the Lord the opposite. All these the shepherd, the angel of repentance. commanded me to write.
Amongst rabbinic writings, Devarim Rabbah, the rabbinic commentary on Deuteronomy, has an angel take on the appearance of Moses.
Devarim Rabbah 2:29, “Bar Kappara said: an angel descended in the likeness of Moses and made him flee and the Egyptians thought that the angel was Moses”;
Kohelet Rabbah, the commentary on Ecclesiastes, has an angel appear looking like Solomon and sitting upon his throne.
Kohelet Rabbah 2:4; “At that time an angel descended in the form of Solomon and sat upon his throne”;
Bereshit Rabbah, the commentary on Genesis, states that Jacob wrestled Esau's guardian angel. Interestingly, it uses sar instead of malach for angel. (The use of sar as a synonym is attested in the ancient writings.)
Bereshit Rabba 77:3 "it is the angel of Esau."
More information can be found in J.H. Moulton, “IT IS HIS ANGEL,” JTS 3516, 519-520 and Strack and Billerbeck 1:781-783; 2:707.