There is a central bank sitting in the middle of most modern economies, and it is a bank-for-banks: they either borrow from it or deposit into it. It sets an all-important interest rate: this rate is the rate-of-return that those other banks will compare possible loans to, when saying 'is it worth it to loan to this person, or should I just put my money in the central bank?' -- if you lower this rate, then presumably banks make more loans, stimulating the economy; if you raise this rate, then presumably banks make fewer, stifling the economy. The hope is not that different from storing up food in years of plenty in order to weather years of famine; during a recession you lower the interest rate, but afterwards in the upswing you raise it again.
When you deposit money in a bank, it is not required to hold onto all of it and keep it safe. Instead it usually takes a chunk of that money and promises it as a loan to someone else, who will hopefully in the end pay them more in interest than they're paying you, enough more that the difference is better than just shoving the money in the central bank.
When it makes this loan, that comes in the form of saying "I still owe you $x, and now I also owe this other person $y." So the true money stored by the bank is still only $x, but the money now in circulation in the economy -- in the sense that there's the illusion of people having it, which is the only sense that money exists anyway -- is $x + $y. If that other person needs to immediately cash out a quantity of dollar bills, then $y < $x, so the bank can comfortably do that. If you both try to cash out at the same time, there is an obvious problem (the bank has failed!), but remember that this is a sort of insurance scheme of "divide that risk among lots of people and trust that their decisions follow a binomial distribution" -- in reality there is some proportion p of a lot of individuals who collectively need to withdraw in order for the bank to fail. Furthermore when this happens the federal government may declare the bank "too big to fail" and funnel tax money into it, to keep it going.
This PDF is arguing that the above explanation is precise in ways that many economics textbooks are loose; notice that the extra value has nothing, for example, to do with your putting the money into the bank. Why? Because suppose you spent that on a new car: your car dealership that you're buying from has an account with this bank (or some other bank; the economy collectively includes all of them after all) and they take this amount of money and put it in the bank! So that $x is getting deposited somewhere, assuming that you do not take it out of circulation yourself (in which case, it doesn't help mediate your purchasing of goods and services, so is it really money?). Modern textbooks omit this and refer instead to the banks as a "money multiplier" where "there is $100 deposited at first, the bank saves 20% of its deposits, it loans out $80 which gets immediately spent hence re-deposited in some similar bank, so that bank saves $16 and loans out $64 which similarly gets re-deposited, saving $13 and loaning out $51, at this point the economy has nominally $100 + $80 + $64 + $51 = $295, so we've multiplied the amount of money deposited into the economy by a factor of ~3x, and if you continue the progression out to infinity you find that this multiplier approaches 1/(20%) = 1/0.2 = 5." The problem with this thinking is that there's nobody in the economy who comes into this system at that crucial step 0 -- you got your money from the bank account of your employer or from a loan from some other bank. So, from your perspective, there's kind of nothing to multiply.
If that makes you feel breathlessly like everything is an illusion resting on an insane Ponzi scheme, that's mostly correct and you should feel all of that. You can spend hours thinking about "what is money, I mean really?" and if you're a founder-mentality you might even spend years on this wondering if there's an opportunity for disruption at an epic scale somewhere therein.
Probably the best way to understand money is still the engineer's, "I do something that someone finds useful, so they give me points in this augmented-reality game where the rules force them to give those points out of their own personal stash. As long as we collectively agree that useful things are happening, those points have some form of meaning as a way to symbolically trade hypothetical goods and services. When we lose this collective confidence then the points will no longer matter to us."