The paradox is that as people attempt to save more, the total amount of savings decreases.
Before I explain how this can be so, we need to understand a couple of things:
First of all, when economists talk about savings in aggregate, they consider the full spectrum from at one end saving a lot, through zero savings (quickly spending exactly what you earn) through to the opposite of saving - which is borrowing. I.e. "borrowing less" can be described as "saving more". Similarly the process of repaying existing loans will be described by economists as "saving more".
Then you need to consider the effects of our "fractional reserve banking" monetary system. In this system loans create money and repaying loans destroys money. This means that if an economy moves to a state where the rate of repayments of existing loans is greater than the rate of taking out new loans then the money supply will shrink (at least without any QE or similar).
Put these facts together and we can have a situation where we have "more saving" (which to an economist may just mean less borrowing), leading to a shrinking money supply. Given that savings are money - then if there is less money around then our ability to save as much as before, diminishes.