First, tax revenue plunges during recessions because more people are unemployed and business activity declines. Consequently, individuals and companies have less tax liability. Also, during a recession, the government spends more money. The government has spent money to strengthen the economic safety net (extending unemployment benefits, etc.). It has also allocated money to rescue the financial system.
Increasing spending while taxes are falling will lead to a budget deficit. In order to pay the bills the government needs to borrow money. The tax cuts passed during the Bush administration has also contributed to declining tax revenue -- and consequently -- to the deficit and debt.
Although the foregoing arguments draw from basic economic concepts, Republicans refuse to acknowledge this fact in public debates. By emphasizing spending instead, they hope to gain political support for cutting social programs to assist the elderly and poor. Surprisingly, Democrats often fail to acknowledge the relationship between the recession and the national debt and deficit, even though that argument could help them rebut the Republican arguments about spending. In sum, both sides are miseducating the public.
For more on this issue, see:
FACT CHECK: Recession is culprit in high US debt (Associated Press)
Fiscal FactCheck (Factcheck.Org)
Debt Ceiling Deal: Preliminary Observations on the Deal AND Obama (Dissenting Justice)