Banks don't lend out your savings. They create new money.
When a bank grants you a €10,000 loan, it doesn't take the money from a vault. It credits €10,000 to your account and simultaneously records a €10,000 claim against you in its books. The deposit is created out of nothing at that moment. When you repay the loan, it disappears again. Most euro money is bank money created this way.
Useful: That's why the money supply grows when lots of loans are taken out and shrinks when they're repaid.
Source: de.wikipedia.org