The answer people believe: it borrows in excess of its taxes.
The way the mechanics actually work: federal government spending (transferred from the Treasury's account at the Fed) in excess of its taxes generates new deposits at banks, and Primary Dealers (determined by and interacting with the NYFED), the 23 largest global banks and financial entities who receive most of those deposits, are required to "make the Treasury market." These 23 large financial entities are required by the FED to ensure Treasury bills sold at its auctions are filled. By law, the federal government has to sell securities in excess of taxes (the Treasury is not allowed to print money, but the Fed can), but its spending in excess of taxes generates the deposits the banking system (the dealers) uses to "fund" those securities. The Fed stands ready to make sure the market functions to meet its interest rate target.
The interest rate on T-Bills is a function of the Fed's target rate. Since the US government's short-term borrowing costs are governed by the Fed, by definition the low Fed target keeps the government's cost down (if this is what you mean by ZIRP helping "government spending", yes), though the low target is a consequence of the sluggish US and world economies.
The key issue, then, is knowing what causes the Fed to change its target and therefore short term T-bill rates...