In order to pay for pensions, the government borrows money from young, working adults. This is effectively what happens in pay-as-you-go public pension systems (which is most of them, to my knowledge, apart from the US, I'm not 100% sure how pensions work in the US). The money you put in actually goes to pay for another person, with the government guaranteeing that they will do the same for you.

If the percentage of retired people increases, the percentage of working adults naturally decreases. Eventually, you'll hit a turning point where the government can no longer borrow from working adults. The government is now in a debt crisis and has to loan money from banks or foreign investors at a significantly higher interest rate, which becomes even more unsustainable if the percentage of retired people increases even more.

This is what is happening in e.g. South Korea and Japan. There are too many old people, and too few working adults. This is caused ny low birth rates over a long period of time.