You can think of it this way, the higher the debt, the more of your tax dollars go towards paying off that debt, which means less money goes into services you think you're paying for. The earners of this are institutions and the wealthy who lend the money to the government.

A government can print more money to pay off that debt (causes inflation), it can lower interest rates on that debt (also causes inflation), cut costs (causes deflation but politically unpopular), or it can raise taxes (also causes deflation but politically unpopular). In the first two cases, the value of your savings diminishes.