During recessions the federal government tries to stimulate the economy (i.e., "prime the pump") by increasing spending and cutting taxes. This requires deficit spending. But state governments have to balance their budgets in both good times and bad. So, to balance their budgets during recessions, states almost always end up cutting spending and/or increasing taxes, thus putting more drag on the economy rather than less.
Since the federal government is responsible for overall macroeconomic management, it makes sense for the federal government to provide fiscal relief to the states during recessions to reduce the amount of budget cutting and tax increasing necessary at the state level. If the federal government doesn't help the states during recessions, then state budget-balancing actions will cancel out a greater portion of the positive impact of federal stimulus efforts.
The "state fiscal relief" that was included in the American Recovery and Reinvestment Act (ARRA) played a major role in allowing New York and the other states to balance their 2009-10 with fewer budget cuts and fewer tax increases than would have been necessary otherwise. Argues that this "state fiscal relief" should be extended so that its phase-out dovetails more closely with the recovery of the 50 states' economies and finances.