$S=I$ is purely definitional. By definition the output (I assume closed economy for simplicity) is given by:
$$Y = C+ I + G $$
saving by definition is equal to net income minus consumption:
as a consequence of the above definitions, which are by definition true identities:
$$S + T +C = C+ I + G$$
and once we solve for $I$ we get:
$$S + T- G = I $$
So by definition investment must be equal to private $S$ and government $T-G$ saving. Many 101 textbooks will for simplicity assume government runs balanced budget so the definition boils down to $I=S$.
However, as a consequence what is considered investment on national account does not correspond to layman understanding of investment. For example, store building an inventory of unsold goods counts as an "inventory investment", similarly for household saving you will have a catch all term called "saving-investment" (see the Measuring the Economy
A Primer on GDP and
the National Income and
Product Accounts published by BEA). Consequently, any required reserves are inconsequential, investment here does not necessarily mean investment in factories.
The leakages and injections have to end up being equal in a given period of time. If the amount of leakages increases then amount of income must raise to offset it. Otherwise $Y=C+I+G$ would be violated.
The leakages and injections are actually based on this expression, which as explained above is purely definitional;
$$S + T = I + G$$
where $S$ and $T$ are leakages and $I$ and $G$ are injections (dont forget we simplified to closed economy maybe your textbook also has $M$ and $X$ there for imports and exports).
If there is an increase in $S$ then simply either:
- Taxes must increase
- Investment must increase
- Government spending must increase
or some combination of the above.