Why is the debate over state versus federal rights (federalism) important?

The debate over federalism is important because it's essential to establish the precise division of powers in such a system of government if it's to work properly. Otherwise, there's a serious risk of conflict between the center and the periphery, with potentially damaging consequences.
One only has to look at what happened during the Civil War to see an illustration of this. The Southern states believed that their right to maintain the institution of slavery was about to be taken from them by the newly elected President Lincoln. In actual fact, they were completely wrong, as Lincoln had no intention of abolishing slavery; he simply wanted to halt its spread. Nevertheless, the fact that the South was prepared to secede over the issue of states' rights (in this case, the right to practice slavery) shows how important it is to maintain a cordial relationship between federal and state governments.
Federalism as a system depends on a smooth relationship between the central government and the states. If that relationship becomes fractured in any way, as it did on the eve of the Civil War, then the whole system is rendered much less effective. Even though the federal government has been very much the dominant partner in the relationship since Reconstruction, it has still been necessary to ensure, wherever possible, the cooperation of the states in the implementation of federal policy.
What has complicated matters over the decades is the vastly increased role of the federal government, which has involved arrogating to itself more areas of government policy that were traditionally the province of the states.
A prime example would be education. Before the Department of Education was established in 1979, education was largely the responsibility of the individual states. But since then, the federal government has become increasingly active in setting strategic policy targets, such as the "No Child Left Behind" initiative under President George W. Bush. Yet at the same time, state governments remain largely responsible for the setting of school budgets, school districting, and the drawing up of curricula. These extensive residual powers over education still enjoyed by state authorities often militate against the attainment of federal policy goals.
It is little wonder, then, that critics have argued that federalism as it now stands has blurred the respective competencies of federal and state governments to such an extent that it has become impossible in many areas to provide coherent policy solutions to a whole range of problems in society. In response, some have argued for an even greater, more hands-on role for the federal government. Others have argued for the precise opposite, advocating a return to the federalism of old, when any powers not specifically attributed to the federal government in the Constitution were deemed the exclusive province of the states.
Either way, a creative tension between the federal government and the states will doubtless remain an integral part of the American system of federalism. How such tension is managed and to what extent it can be turned to good use is a different matter entirely.