This is quite a complicated subject, maybe not in a technical sense, but it takes some time to wrap your head around it. I will try to do my best to give a basic answer, and maybe someone else can give more details.
Varoufakis could be referring to one or two different but related things.
The first thing is that the modern monetary system is based on so-called Fiat money (or paper money). Up until 1971 and the collapse of the Bretton Woods system, the US dollar and other Western world currencies were backed by a gold standard, e.g. US dollars were legal tender whose value was directly comparable to the value of a certain amount of gold. The advantage of such a system was that the amount of gold in the world put a limit on how much the value of a dollar could be diluted by inflation (printing more money). For reasons I will not develop further here, the Bretton Woods system fell apart following the Vietnam War and the energy crisis of the 1970s. Today, there is no physical object backing up the value of currency.
Instead, trust in the economic institutions are what gives money its value. As long as people are willing to exchange goods, services and securities for money, the system works. When the financial crisis struck in 2008, the world-wide banking system was on the verge of a situation where people became afraid to engage in economic activities, because if trust dissipates, in a crisis scenario, everything stops in the economy and people can no longer transfer their funds, have their wages paid, buy goods with their credit cards, etc.
Because currency is Fiat money, central banks can “print” money at will. Or in reality, what's happening is that the central bank tells its computer network to make a big number with a bunch of trailing zeroes, and then use that money to buy assets from the market. That way, they can give banks and other financial institutions access to "fresh cash". Of course, in the long term, this is merely an artificial boost that dilutes the value of each currency unit in the total money stock. But it has a short-term stimulating effect on the economy.
The other part of this is that the financial system is based on what's known as Fractional Reserve Banking. There are good videos and articles explaining how it works. But in a nutshell it means that: when a creditor deposits funds into a bank account at Bank A, the bank is only required to keep a fraction of those funds in the bank. The rest can be used for proprietary investments, lending out to other customers, etc. When that happens, those funds are often effectively just being deposited into another bank account at Bank B. This means that Bank B just needs to keep a fraction of the deposits, and can lend out the rest which lands in the books of Bank C. This repeats ad infinitum (as far as I recall), but since a fraction is taken away all the time, in the end there will be a limit on how many times a single unit of currency can be multiplied.
The advantage of the Fractional Reserve System, as I understand it, is that if the bank has to hold a small reserve requirement, the expansion of economic activity is much greater than if a bank had to hold a large fraction in reserves. Of course, if the bank had to hold 100 % of the deposit available for withdrawal at any time, there wouldn't be much lending going on at all. The problem is when the investments are badly managed, or if there is a bank run and everybody wants to withdraw their money at the same time. This was one of the fears in the 2008 financial crisis. At that point, those deposited funds on your bank account are not really there, because they are invested in burgeoning investments whose dividends may not have materialized yet. Or even worse, they are invested in asset bubbles which may burst at any time, which happened in the subprime mortgage crisis of 2007-2008.
Thirdly, the combination of Fiat Money and Fractional Reserve Banking means that when the bank allocates you a loan so you can build a house, that doesn't necessarily mean that they have the money in reality. In some sense, they are just creating a bunch of digits in your bank account and count on that the real value created by the building of the house will be able to cover those liabilities once finished.
So Varoufakis statements are a sort of criticism of the modern monetary system, with good reason. However, as long as the system works, it serves its purpose by accelerating economic growth in a higher rate than would have been possible in a more constrained system. But the backside is that there will always be booms and busts in such a system and great risks if greed triumphs and trust dissipates.