Earlier quoted context omitted.
Private banks increase money supply by lending. If 10 people deposit $1000 in a bank, it can loan $9000 to an 11th person. Now the economy has $19000 total.
The $9000 has to be paid back, and then some. I sure hope you aren't an accountant.
Now to expand GP's example (still simplified):
- A borrows $100k money to pay B toward building a house. B puts $100k in their bank.
- C borrows $90k from B's bank toward building a house to pay D. D puts $90k in their bank.
- etc
So, houses were created (or other services were provided), and that's the real multiplicative factor. If banks loan out 90% of the cash stored (i.e. keep 10% in reserve [1]), the multiplicative factor of value creation in the economy is 10x the original amount of cash deposited in the first bank.
Now, if all of us withdrew our savings at once or sold all our stocks at once, we would have an economic shock analogous to that which resulted the Great Depression. That's why for banks, we have FDIC insurance - to mitigate such a panic so that money can serve its value-multiplicative role when it's not being actively used for anything else by the person owning the money. That's also why a positive (but low) inflation was originally considered economically healthy - so that people put their money in banks/market rather than under their mattresses gradually losing value. When interest rates are low, that encourages people to put their money into riskier (non-FDIC-insured) investments with higher growth potential, like a balanced portfolio of stocks/bonds/etc to avoid losing value to inflation, resulting in more economic growth.
[1]: https://en.wikipedia.org/wiki/Fractional-reserve_banking