I recommend you read "Blessed: A history of the American Prosperity Gospel" by Kate Bowler if this topic is of more interest to you, but I'll provide an answer for you below.

Short answer: the Bible can support almost any position if you curate your proof texts appropriately.

Long answer:

The prosperity gospel focuses on obtaining "God's blessings". While not necessarily in line with what Jesus taught his disciples, or what his disciples taught the early Christian church ("no one claimed that any of their possessions were their own, but they shared everything they had." -Acts 4:32), there are plenty of examples that can be found in the Old Testament to serve as proof texts for those wanting to believe in the idea that following God leads to earthly wealth.

Abram, later known as Abraham, is traditionally a wealthy man with large herds, servants, etc. ( "Abram had become very wealthy in livestock, silver, and gold." -Genesis 13:2) His wealth is extensive enough that eventually he and his relative, Lot, have to part ways because their herds are too large to share the same pasture land. ("But the land could not support them together, for their possessions were so great that they were not able to stay together." - Genesis 13:5)

Later in Genesis chapter 13, God essentially pulls a Mufasa and tells Abram to turn around in a circle, and everything he sees will be his.

David, the shepherd boy turned king, constantly speaks of God's blessings and how God has blessed him through the years. His success is seen as a sign of God's favor in his life. (Samuel chapter 2)

Solomon's wealth is a direct blessing from God as is his wisdom both of which come from Solomon offering any enormous sacrifice before God and then summarily being given great power and wealth as a result. (1 Kings 3:10-14)

Even Job, the poster child for suffering without reason, is blessed by God for remaining faithful and is blessed in a financial way.

"After Job had prayed for his friend, the Lord restored his fortunes and gave him twice as much as he had before." - Job 42:10

All of these figures in the Bible are used to prop up the idea that God will make wealthy those whom he favors. There are even passages in the New Testament that can be used to support this claim. One example being Luke 6:38 -- "give, and it will be given to you. A good measure, pressed down, shaken together and running over. For with the measure you use, it will be measured to you."

There have even been (erroneous) claims put forth in Christian circles that the "eye of a camel" was referring to a gate on Jerusalem known as the camel gate, so Jesus was stating that it was merely difficult, not impossible for a rich person to be saved.

Over time, many of these preacher's also adopted Jesus' parable of the sower (Matthew 13) towards their own purposes, stating that "sowing a seed" (usually a tithe, offering, or donation to the specific church) would entice God to bless them a hundred times over (forgive their debts, heal their sickness, etc.)

Prosperity Gospel preachers use all of this to prop up their claims, and then once they become successful and wealthy themselves, can point to their own success as an example of God's blessing being manifested in their own lives. (Creflo Dollar, and Kenneth Copeland are masters at this technique)

As the Bible is a highly symbolic text shrouded in cultural mores often unfamiliar to a 21st century audience, it is fairly simple for someone (not just a prosperity gospel preacher) to twist the bible to their own needs and make a convincing argument while doing so.

We've seen this throughout history, not only with the protestant reformation and counter reformation, but even in more modern times with the hundreds of Christian groups that all claim to interpret the Bible in the correct way, ranging from Mormons, to the Church of Christ Scientist, to more obscure movements like Theosophy. Not to say any of these groups are wrong, but they all interpret scripture in wildly different ways.

While this isn't a historical analysis, per say, I want to draw your attention to the idea that every Christian group would consider themselves "well versed" in scripture, and the myriad of interpretations speaks to the difficulty in understanding a collection of texts written over thousands of years by tens of authors, before being translated through several languages to reach a modern congregation.

Even theological ideas taken as gospel (pun intended) by most groups like the doctrine of the Trinity, or the idea of Heaven are not explicitly stated in the Bible.

Edit: typos.