Alan Horowitz wrote:
"---how does the voltage "know" that it should be increasing exactly 63%
during each time-constant period?"

The rate of growth or decline is natural as the circuit response is
non-variable and operates on the energy flow in the circuit at the
instant. This rate depends on the state of charge in the capacitor while
the capacitor is charging. Rate of capacitor discharge depends on the
charge remaining in the capacitor. It is steadily (exponentially)
declining during discharge.

The time required to charge a capacitor to 63% (actually 63.2%) of full
charge, or to discharge it to 37% (actually 36.8%) of its initial charge
or voltage is defined as the "time constant" of the circuit.

A search on "time constant" will produce many colorful illustrations.

Time constant is the time in seconds for a capacitor to charge up to 63%
of the applied voltage, or the time it takes a fully charged capacitor
to discharge from 100% down to 37% of full charge.

Time constant is the product of R (in ohms) times C (in farads) in an RC
circuit.

Time constant is the quotient of L/R with L in henries and R in ohms in
an RL circuit.

Epsilon is a number approximately 2.71828 which is the base of the
natural, Naperian, or hyperbolic logarithms.

There is a natural rate of growth or decline caused by growth or decline
as a constant percentage of size at the moment. It is epsilon.

As an example of natural decline, the quantity of charge (q) remaining
in a capacitor after current has been flowing out for (t) seconds is:

Qo times epsilon raised to the minus t/CR power.

Where Qo is the initial value of q.

Voltage is proportional to charge, so V at any time can be found by
substituting Vo for Qo in the formula for q.

Best regards, Richard Harrison, KB5WZI