What are the widescale economic ramifications of people holding onto their money instead of spending it? Pretty simple question: people get rich by saving more money than they spend. What are the implications of people holding onto their money instead of spending it?

Do rich people save more money (ratio-wise) than less rich people?
The amount that people save depend on the marginal propensity to consume (MPC), i.e., part of the additional income that a person consume, hence, $1-MPC = savingrate$. Keynes assumed that with increase in income MPC would increase, while neo-Keynesian assumed it to be constant in their analysis. But real data shows that MPC increases with elevation of income but decreases with large income gains. Hence, in a way, rich people save more relatively to others. But that is not the reason why they are rich, more saving is the effect, not the cause. Mostly, the reasons behind getting rich are the apt investment decisions.

When an economy hold more money, consumption reduces leading to reduced outputs. This in return decreases real income. Since, in equilibrium savings equals investment ($I = S$), so actually there is no increase in savings and we are left poorer with lower income levels. This is called the paradox of thrift, suggested by Keynes.