Think about it this way. Imagine your household income is $50k, the median. But for one year, you decide to live like you are at $30k, barely above poverty. It may be a difficult year but there is no reason you can't manage, 1 in 5 people live their entire life with that. At the end of the year, you are left with $20k. You can now start living your life like anyone with a median income will. Because of that 1 frugal year, you are left with a average balance of $20k for your entire life. No more living paycheck to paycheck. And even better, when you die, your children will inherit these $20k so they won't need that year of poverty to keep their balance positive. So these $20k will actually last forever.
Of course, it is a simplification, but the idea is that keeping savings is just a matter of living with the paycheck of a few months ago instead of the last one.
Now what are the benefits of having an average of $20k that you won't touch.
First, it is insurance. Insurance costs you money in the long run, that's how insurance companies pay their employees. That sum is your own insurance for small things, like phones.
Second is that you can take advantage of opportunities, like volume discounts. Money will make you save money.
For those who are above the poverty line, the reason they can't save is lack of discipline. Not "the 1%". But to be fair, society in North America doesn't really help, with ads everywhere and credit everything.