This answer is from the perspective of the theology of Emanuel Swedenborg (1688–1772), and of the various denominations that accept his theology, commonly called "New Church" or "Swedenborgian" denominations.
Swedenborg taught that there are no angels pre-created as a separate race, nor did any such pre-created angels fall from heaven to become Satan and his angels. Instead, he taught that all angels and demons were once human beings who lived in the material world, and that Satan or the Devil is a collective term for hell.
As explained more fully in my answer here, Swedenborg is the primary source of the modern-day belief that people become angels after death. Swedenborg himself, though, saw this idea as firmly based on the Bible's depiction of angels.
This answer draws heavily on the article, What the Bible Says: Where Angels Come From, which is also written from the perspective of Swedenborg's theology.
The Bible says nothing about angels being created
Genesis 1 describes the creation of "the heavens and the earth," and everything in them. And yet, in the Creation story there is no mention of God creating angels. It seems unlikely that God would leave such an important created being out of the Creation story.
The creation of angels is not mentioned anywhere else in the Bible, either.
The passage about "Lucifer" in Isaiah 14:12 is often cited as evidence for pre-existing angels. However, this prophecy was not interpreted as being about a fallen angel until centuries after it was written. In the original Hebrew, the word that is translated "Lucifer" in older and more traditional translations actually refers to the morning star (i.e., Venus), using it as a metaphor for the king of Babylon.
Though there are non-Biblical texts describing angels as a separate race of beings, the Bible itself never says that they are a separate race, nor does it make any clear statements about where they come from. The idea that angels are a separate race is based more on tradition and church doctrine than on the Bible.
And as covered in the remaining points, there are many indications in the Bible that angels are indeed humans who have gone on to heaven.
The Bible often refers to angels as "men"
The Bible commonly uses the word "man"—Hebrew אִישׁ ('iysh), "man"; Greek ἀνήρ (anēr) "man" or νεανίσκος (neaniskos), "young man"—to refer to angels.
Here are some examples:
While I was speaking in prayer, the man Gabriel, whom I had seen
before in a vision, came to me in swift flight at the time of the
evening sacrifice. (Daniel 9:21)
Here Gabriel, who is seen by Christians as an angel based on Luke 1:19, 26, is called a "man."
The two angels (see Genesis 19:1, 15) who visited Lot in Genesis 19:1–29 are called "men" in Genesis 18:2, 16, 22; 19:10, 12, 16, and in several other verses in Genesis 18 & 19.
When an angel appeared to Manoah and his wife (Samson's parents) in Judges 13, they identified him as a "man of God," and both they and the narrator also refer to him as a "man"—although he is also clearly identified as an angel. Manoah even calls the angel "a man" when he talks to him (see Judges 13:11), and the angel does not correct him. This suggests that the angel had no problem thinking of himself as a human being like Manoah.
In Zechariah's vision of the horsemen in Zechariah 1:7–17, the lead horseman, who was "standing among the myrtle trees" is referred to both as a "man" and as an "angel."
The angel(s) at Jesus' empty tomb is(are) referred to as an angel in Matthew 28:2–7 and as two angels in John 20:11–13, but as a man in Mark 16:5–7 and two men in Luke 24:4–8.
In short, the Bible uses "angel" and "man" almost interchangeably when speaking about angels. If angels were a separate race, the Bible would not refer to them as men, or people. That would be like calling a horse a sheep, or a pig a donkey.
The fact that there is no separate word for "angel" in the Bible (see below), and that angels are commonly referred to as "men," strongly suggests that angels are humans.
Angels look like people
As seen in the passages quoted and referred to just above, when people on earth encounter angels, they commonly think that they are meeting a human being rather than an angel. The writer of the letter to the Hebrews even says:
Do not neglect to show hospitality to strangers, for by doing that
some have entertained angels without knowing it. (Hebrews 13:2)
Although angels are sometimes described in the Bible as having shining faces and clothes, this is not unique to angels. Moses' face also shone after he had spoken with God (see Exodus 34:29–34). And in the final verse of the Parable of the Weeds among the Wheat in Matthew 13:24–30; 36–43, Jesus says:
Then the righteous will shine like the sun in the kingdom of their
Father. (Matthew 13:43)
So Jesus himself says that people who are righteous will shine in God's kingdom—just like the angels.
"Angel" means "messenger," not a separate race of beings
In both the Hebrew of the Old Testament and the Greek of the New Testament, the word for "angel" simply means "messenger." It does not refer to a separate race of beings. The Hebrew word is מַלְאָךְ (mal'ak), "messenger"; the Greek word is ἄγγελος (aggelos), "messenger."
Although angels are sometimes described as great, holy, powerful, and so on, the same descriptions are also used for people. For example, both angels and people are called:
In fact, human beings are also called "messengers" many times in the Bible, using the very same words, both in Hebrew and in Greek, that are used of angel messengers. For just a few examples, see Genesis 32:3; Deuteronomy 2:26; Joshua 6:17; Matthew 11:10; Luke 7:24; James 2:25.
In short, "angels" or "messengers" in the Bible can be either human beings on earth or angelic beings in heaven. Both of them are described using the same words.
Angels themselves reject the idea that they are superior beings
Not once, but twice in the book of Revelation John fell down to worship at the feet of the angel who was speaking to him. Both times the angel stopped him, making himself equal to human beings under God:
At this I fell at his feet to worship him. But he said to me, "Don't
do that! I am a fellow servant with you and with your brothers and
sisters who hold to the testimony of Jesus. Worship God! For it is the
Spirit of prophecy who bears testimony to Jesus." (Revelation 19:10)
I, John, am the one who heard and saw these things. And when I had
heard and seen them, I fell down to worship at the feet of the angel
who had been showing them to me. But he said to me, "Don't do that! I
am a fellow servant with you and with your fellow prophets and with
all who keep the words of this scroll. Worship God!" (Revelation
Jesus says that we become like angels after death
In the incident of the question from the Sadducees about the Resurrection, recorded in the three Synoptic Gospels, Jesus says we will be "like" or "equal to" the angels. See Matthew 22:30; Mark 12:25; Luke 20:35–36. And the saved are described as having powers similar to those that angels might wield. See, for example, Mark 16:17,18; 11:23; Luke 10:17,19; John 14:12.
Yes, it could be objected that just because people can be just like angels, that doesn't necessarily mean that they can become and be angels. But if it looks like an angel, walks like an angel, and quacks like an angel . . . .
Nowhere in the Bible does it say that angels are a separately created race of beings.
Everywhere in the Bible, angels are described as being human, as having characteristics and powers similar to those possessed by great human beings, and as engaging tasks similar to those that human beings engage in.
Further, both angels and Jesus himself make humans—especially humans who have died and been resurrected—equal to and in every way like angels.
The main difference between humans and angels is that humans live on earth while angels live in heaven. Other than that, it's difficult if not impossible to find any real differences between them. And since it is commonly believed by many, if not most, Christians that we will live in heaven after death, that would leave no differences at all between angels and humans.
Why would God create two different races of beings that look the same, speak the same, think the same, and act the same as each other?
Based on all of this, it is quite reasonable to conclude based on the Bible's stories about angels that they are indeed human beings who have gone on to the spiritual world and become angels. After all, Jesus himself said to one of the thieves on the cross:
Truly I tell you, today you will be with me in paradise. (Luke 23:43)
Once again, this answer is given from the perspective of the theology of Emanuel Swedenborg, and of the New Church or Swedenborgian denominations that accept his doctrines.
These church denominations all hold the view that there are no angels created separately, but that all angels were once people who lived in physical world, and after death went on to become angels in heaven.