How did that belief change into the modern view that people who die and go to heaven will stay in heaven eternally?

Both beliefs have existed simultaneously since ancient times and survive today. Let's start with a look into the origin of the former. When the Israelites were exiled from Palestine by Babylon in the 6th century BCE, returning to that land became a major theme in their scriptures. This is the purpose of Ezekiel's vision of the Valley Of Dry Bones, in which God shows the prophet many dead bodies being returned to their living state and explains that Israel will be restored just so.

After the exile, many exilic biblical passages were reinterpreted as Israel's historical context changed, and in the face of oppression many Jews began to view Ezekiel's vision as a prophecy of a literal, bodily resurrection at the end of time, divine justice for the suffering they faced in life. We see this view in the book of 2 Maccabees (which FYI isn't considered canon by protestants) written in the second century BCE. The author recounts the oppression the Jews faced under the brutal Antiochus Epiphanes, who outlawed Jewish practices under pain of death. In the story, a mother and her seven sons bravely choose to be burned alive rather than renounce their faith, comforted by the knowledge that they will be resurrected to eternal life while Antiochus and his men rot in the ground.

A few centuries later we see a retelling of the same event in the book of 4 Maccabees, but this time the family is reassured not that they will be bodily resurrected, but that their souls will be immediately gathered to God, and that

We, through this severe suffering and endurance, shall have the prize of virtue and shall be with God, on whose account we suffer; but you, because of your bloodthirstiness toward us, will deservedly undergo from the divine justice eternal torment by fire. (4 Mac 9:8-9)

For this author it is the soul that matters rather than the body, and since God is eternal, so is his judgment, immediate and everlasting. (The irony of the influence of Platonic philosophy on this author's account of a resistance against Greek culture is apparently lost on him.)

The persecution under Antiochus is the same situation that gave rise to apocalyptic literature like the book of Daniel (2nd century BCE), in which the author assures his reader that God will send "one like a son of man" to bring divine destruction onto the ruler and usher in the bodily resurrection and judgment immediately after his death (Daniel 11:45-12:2). The early Christians adapted this framework to Jesus. They believed he was Daniel's "son of man" who intervened to conquer death and sin and that his resurrection was the "first fruits" of the general resurrection soon to come (1 Corinthians 15:20).

Many believers changed their minds from belief in the resurrection to the belief in immediate spiritual judgment simply because God's physical intervention into history (known as the Kingdom Of God) and ushering in of the resurrection never came. We can see this even in the gospel of Luke, when Jesus tells the man being crucified next to him "today you will be with me in Paradise." The evangelist even tells his readers not to expect the Kingdom Of God to come, but rather know that it resides inside them (Luke 17:21). Compare this to the other gospels in which Jesus assures his followers that the earthly Kingdom Of God is soon to come indeed (Matthew 24:30-34).

Plenty of Christians today believe in a synthesis of both immediate spiritual judgment and a bodily resurrection at the end of time. This belief goes back to the book of 4 Ezra, in which the eponymous Israelite leader is told by an angel that spiritual torment and paradise begin immediately after death, but that there is also a coming bodily resurrection, at which point the righteous will enjoy everlasting life and the wicked will finally be destroyed forever.

Source: Dr. Bart Ehrman's Heaven and Hell: A History of the Afterlife

A lot could be added to address shifts that have happened in the past five hundred years, but /u/AndrewSshi's responses to a question about the origin of indulgences as a practice covers how beliefs around heaven developed in the first fifteen hundred years of Christianity. That post is here.