Basically there's diminishing returns on how much additional income improves your life. Your first few thousand dollars of income are crucial, giving you food and shelter. The next few thousand are nice, making your life better with cell phone, better housing, basic medical care, etc. The difference between $1million and $1.5 million gets you nicer things, sure, but your standard of living doesn't improve nearly as much as it did from $10,000 to $15,000. Once you're making $100 million/yr, another million/yr is appealing, especially as a symbol of how much people value you, but it doesn't really make much difference in how you live your life.
Now assume there are some crucial functions of government that cost X - firefighters, police, roads, food safety, whatever you consider legitimate uses of government. You have to design a way to get X from the populace in the way that encourages growth while hurting the economy and the people the least.
Say you set your exemption low, say $15k/yr. That excludes about 13.5% of the population. Here's a tool to estimate what rate of taxation you'd need with a flat tax:
http://politicalcalculations.blogspot.com/2011/09/design-your-own-flat-income-tax.html#.UYha3cof-KI Last year tax revenue was $1.1 trillion, which comes out to about a 25% flat tax.
Everyone has to pay enough to make up that exemption, so you actually need to charge proportionately more, which for someone making $20k/yr really really hurts (down to $15k), but for someone making $10 million/yr is no problem (still has $7.5 million). That means you're making it easier for the rich to stay rich and making it harder for people to struggle out of poverty. You're not encouraging growth, and while you're being "fair" in the sense that everyone is paying the same percentage (edit: has written 'number' here) of dollars, you're not being "fair" in the sense of spreading around the
pain of taxation as lightly as possible.
If your exemption is more generous, say $40k, you're now asking just 60% of America to pay the taxes for everyone, which 1) will create resentment among the 60% and 2) makes it much more beneficial to report $39k income since you join the taxpayers at $40k and get a big tax burden. You're creating a big hurdle to people trying to advance and a disincentive to earn more money unless you can earn a LOT more money, which is unlikely for most people.
With progressive taxation, you're trying to make everyone "feel the pain" of taxes equally, while always creating incentive to earn more. There's no point in a progressive tax system where a higher salary + new taxation on it leaves you with less money - $100k after taxation is still more than $99k after taxation, and that's true for any increase. The guy making $100million/yr has to pay more dollars, but that last $20 million/yr wasn't going to be necessary to send him to college, buy his groceries and healthcare to keep productive in the workforce, or otherwise be as big of a boost in making him productive in society and his life significantly better like it would for the poor.
One thing lots of people on the far right (Milton Friedman) and far left agree about is actually having a
negative income tax bracket at the bottom so that you keep this incentive system of always getting more as you work for it, but avoiding the nanny-state micromanaging of direct welfare payments.
So long story short, a flat tax with exemptions doesn't hurt those below the exemption, unless to make up for the lost revenue the government finds a way to tax them anyway, like DMV fees, road tolls, etc. but it does hurt their chances of building their way to the top through hard work and talent. It REALLY hurt those right above the exemption.