Simply put, it means that one cannot get something for nothing. Even if something appears to be free, there is always a catch. You may get free food at a bar during "happy hour," but the bar-owner either figures out a way to get you to pay or gets some sort of benefit (such as attracting new customers). This may or may not be true at the individual level, depending on the interpretation of the phrase.
Though it is possible for an individual to get a "free lunch" (as when a company cuts its costs and gains competitive advantage by polluting the air), someone ends up paying the cost of the "lunch." Even though there is no individual or private cost, there is a Social cost. Similarly, someone can benefit for free from a beneficial Externality or from a Public good. But someone has to pay the cost of producing these benefits.
Strictly speaking, the idea that there is no free lunch at the societal level applies only when all resources are being used completely and appropriately, i.e., when efficiency prevails. But when Inefficiency exists, one can get a "free lunch" by abolishing it. For example, microeconomics argues that the pollution example of the previous paragraph is allocatively inefficient. A tax or other program that forces the polluter to internalize this externality would improve efficiency, increasing social welfare. However, others may be benefiting from the inefficiency and use their political or social power to prevent you from doing so. That is, the polluter may use lobbying and campaign contributions to preserve his or her ability to legally pollute.
"A sperm-and-ova bank insures the future of the race; a tissue bank insures the future of the individual. But somebody has to pick up the check; it's a tanstaafl situation." -Lazarus Long in The Number of the Beast, chapter 44. To a scientist, tanstaafl means that the system is ultimately closed - there's no magic source of matter, energy, light, or indeed lunch, that cannot be eventually exhausted - meaning no more lunch for anyone!
Advocates of the TANSTAAFL principle believe markets are efficient unless due to interference by the government or other "outside" forces. The Free market is seen as the solution to issues such as pollution.
The TANSTAAFL argument may also be applied to natural physical processes; see Thermodynamics. In mathematical finance, the term is also used as an informal synonym for the principle of no-Arbitrage.