I was recently asked why I believe in minimum wage laws, since they seem to violate free-market principles. The short answer is that the labor market is not a completely free market. Workers cannot simply change jobs whenever and wherever they want. Moving to a new location is difficult and costly. Workers with two jobs who want to leave one of them have to find a new job that fits into their same schedule. People who live paycheck-to-paycheck will be unwilling face the uncertainty of being "between jobs" for any significant length of time.
The labor market thus has some elements of a free market and some elements of a captive market. It is thus possible for businesses to take advantage of this inertia in the labor market and reduce wages to a level that is far below the value of the employees' work. In a worst-case scenario (e.g. a city with very high unemployment), some companies could theoretically reduce wages to starvation levels in order to maximize profits. We need a minimum wage to guard against this possibility. The United States' inability to enforce its immigration laws only adds to this problem because illegal immigrants can flood the labor market.
When the United States transitioned from a society of independent farmers to an industrial society, we made an important trade-off. While we gained efficiency, wealth, and longer lives, we ceded some control of our individual livelihoods to corporations. Because of this loss of control, we need measures in place to prevent corporations from exploiting individuals.
At the same time, if minimum wage laws force companies to pay employees more than the value of their work, those employees will not be hired. Clearly this is another scenario that needs to be avoided as much as possible. Employers do not have a duty to pay their employees a "living wage" (an extremely ambiguous concept in itself). That is why the minimum wage needs to be kept low so as to not drive up unemployment, but cannot be eliminated altogether.