The difference between "rich people" and everyone else is that while ordinary people have income and own assets the rich have "capital". Capital is a productive asset which returns a yield. Now even low yielding capital such as FTSE100 shares have an average yield of 4% (on top of which the owner may get capital growth) while the economy is growing in the 2-3% range.
This means that provided the yield is re-invested into capital the capital owner will own an ever increasing share of the economy with every passing year.
He could even skim off 1% and still increase his share of the pie.
That is why the rich get richer.