Starve the Beast is an American euphemism for Don't pay for Government.
This euphemism is commonly used by Neo-Conservatives and the Republican Party. Neo-Conservatives and the Republican Party oppose the policies of the New Deal, progressive taxation, public property, and a Welfare State, which they refer to as Big Government. Since most of these policies and programs are very popular, they have developed euphemisms for their preferred policies.
They don't dare come right out and support cutting popular programs like Social Security or Medicare or farm subsidies or the Land Bank, Consumer Protection, Environmental Protection, Financial Regulation, Business Regulation, etc. Conservatives and Republicans favor lowering Taxes (especially for themselves and their country club friends) and know that eventually the debts run up by unfunded programs will force those programs to be either reformed or abandoned. They also don't wish to be seen to have abandoned the prudent fiscal principles of traditional conservatism.
So, Starve the Beast is Neo-Conservative code for cut the taxes that fund Government, and Republican Party policy is to run huge deficits so that eventually the New Deal will fail if Republicans are in power, and scream about debt and the deficit like prostitutes asked to work on credit when Democrats are in power.
Starve the Beast dates from the Reagan Administration, and the deficit run up after all these years, two major banking crises, several unfunded wars and the present economic malaise is enormous. The yearly payment for interest on the accumulated debt is now a large part of every budget.