In short (and this is ripped almost straight from The Deficit Myth):

1. Propaganda about the deficit is presented as "we have to Tax And Borrow before we can Spend anything" (TABS, for short). It's equating national spending to your personal spending, essentially (something most people will agree is silly even if they don't fully understand why).
2. What's the #1 reason your personal spending isn't the same as national spending? You don't have a money printer. If your expenses outpace your personal income, you have to borrow money to cover those expenses. If the federal government's expenses outpace the income it brings in via taxation, it can borrow to cover those expenses -- or -- it can print more money. And this is functionally how the federal government actually operates; it's not some thought experiment or hypothetical. There's an example in the book of Congress approving billions more in defense spending, no taxes or borrowing being arranged to cover it, and the federal government just printing more money to fill the gap. Reality is closer to Spend, then Tax And Borrow (STAB, for short).
3. So can we just print infinite money and do fully-automated luxury gay space communism tomorrow? No, for two big reasons. First, there are still real resource and labor constraints that apply even if we have all the money in the world. Say you want to buy all the yachts in the world. There are only so many now, and if you want more you have to build them, and if you want enough more you eventually have to re-tool a bunch of the economy to continue building them, etc. And because it eventually gets harder to build more, it gets more expensive to build more, which brings us to the second reason we can't just print infinite money and achieve utopia: inflation. If you print too much money but you're not increasing real-life production at a similar pace, you have more and more dollars chasing around a similar amount of real goods. This drives up the price of those goods, and if this gets out of control (for example, if you printed infinite money) it creates all sorts of larger economic problems.
4. What about Vuvuzela? And Greece? What I described in Point 2 only works if (a) your government owns a money printer and (b) the debts it owes can be paid with the type of money that printer prints. Greece lacks the former, as it adopted the Euro and doesn't have the authority to print Euros. Vuvuzela lacks the latter, as a lot of the debts they have to pay must be paid in foreign (to them) currencies. The U.S. (and a number of other advanced economies) have both. Note also that this explains why individual U.S. state budgets don't work the same way the U.S. federal budget does: states (and smaller political subdivisions) can't print their own currency.
5. Wait, why even have taxes at all, then? Taxes serve a number of functions even if they aren't necessary to finance government spending. They can be used to combat wealth inequality (which is corrosive to democracy), they can be used as incentives, and they can be used to restrict the supply of money in the economy to help curb inflation.
6. OK, so we can deficit spend without it really being a big deal. Are we anywhere close to the point where we'll be spending so much it will produce inflation and those larger economic problems mentioned in Point 3? Probably not. We're running historic deficits but inflation is also low. It's likely we can do a lot more deficit spending before it becomes a problem, and we almost certainly will do so -- the question is what we're going to spend that money on.

Would you like to know more? Here's a Citations Needed episode on deficit mythology where they interview the author of that book.