"It is not very unreasonable that the rich should contribute to the public expense, not only in proportion to their revenue, but something more than in that proportion." -- Adam Smith
Taxes are the price of living in modern society with public goods. Many, if not most, agree that taxes are required but dislike the convoluted tax forms that place a burden on businesses and people.
The complexity of the tax codes has given rise to a whole industry of intelligent professionals who will ensure your taxes are correctly done, and often will help you game the system to reduce your tax burden. The complexity also leads to unintended interactions between rules, for example the Clinton 1997 tax break on housing capital gains is thought to have helped drive the housing bubble - in hindsight a tax haven was created that provided incentive to "flip" houses and inflate the bubble. The current situation is one where the average citizen has a strong aversion to taxes, everyone has their favorite anecdote of the rich manipulating their way out of paying taxes, the transaction costs on small businesses are sizable, and resources are unproductively used by providing incentives for intelligent people to work in the "gaming" industry. The flat tax proposes to solve these problems.
The key attraction of a flat tax is its simplicity, reducing unintended effects and conflicts by sweeping away the complexity. Everyone takes their income and multiplies it by the flat tax rate. Every business does the same. Simple. Easy. Transaction costs are negligible, while unexpected distortions and gaming are largely eliminated. Many nations have moved to a flat tax system, and the results are promising (see The Flat Tax by Robert E. Hall and Alvin Rabushka).
The feature most disliked about the concept is its bluntness - incentive tuning cannot be done, for example by promoting certain behavior such as charity donations, and the tax is not progressive. As money has more value to those on the lower income distribution most - including Adam Smith, see the introductory quote - see progressive taxes as more "fair": at high incomes substitution and quality is the main gain to marginal increase in income, at lower incomes crossing a threshold from doing without to being able to obtain goods is the outcome of marginal increases.
The difference between not having and having is much more stark than the difference between good and better. In addition those at the higher incomes often received much of their gains by luck of birth and by social programs, such as subsidized higher education, which also informs peoples preference for progressive taxes. Proponents of flax taxes attempt to take this into account by including an income exception that is removed from earned income before applying the flat tax rate: at the lowest end this removal can either be zeroed or be used to apply negative tax (i.e. returns, or "automatic welfare", such as proposed by Milton Friedman). The fact that this feature is included speaks to the desirability for a nonflat aspect: what is important about "flat" taxes is not flatness per see (although some do think that flatness is the crucial feature) but instead the narrowness of the tax structure. Consider a tax system streamlined to a single parameter, income, multiplied by a single variable - a progressive tax rate. The crucial gains of flatness are gained - reduced transaction costs, eliminating unjust gaming of the system, decreased social disillusionment, and freeing of human capital to productive ends - while the progressive feature, which is already incorporated in flat tax proposals in an attenuated manner (via the exception), is retained.
It is important to note that Hong Kong, which consistently ranks among the top economic performers, has actually implemented such a system over 50 years ago and retains it; Powerful support for simplicity that retains progressive features. Hong Kong also allows deductions to incentivize charitable giving and house ownership (see "Hong Kong's Excellent Taxes", by Alan Reynolds at Townhall.com).
When it comes to taxes, perhaps it is time to heed the words of Matthew and join Hong Kong: "Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life, and only a few find it."