The advantages of a flat tax system are to a large degree in the eye of the beholder. That is, not everyone can agree on what those advantages are.
One advantage that is claimed by flat tax proponents is that such a system would lead to more economic growth. It would remove tax laws that distort economic choices. People would use their money in ways that would be most economically beneficial, not in ways that would allow them to avoid taxes. This would lead to more growth.
A major advantage claimed by proponents is that a flat tax would be fairer. Of course, this is not an objective standard. Proponents argue that it is unfair for people with more money to pay higher percentages of their income in taxes. A flat tax would treat everyone the same.
There is, however, one advantage that is indisputable. That is the fact that a flat tax would be much simpler to administer. Our current tax code is amazingly complicated, making it very hard for people to fill their taxes correctly. A flat tax would simplify things and save people a great deal of time and money.
These are the advantages most commonly claimed for a flat tax.