Alan Horowitz wrote:

when a current just starts flowing into a RL or RC circuit, how does
the voltage "know" that it should be increasing exactly 63% during
each time-constant period?

And whence the number 63%?

This all goes back to the solution of the differential equation for
the RC or RL system. e is a natural constant that has some very sweet
properties in many applications of mathematics, and simplifying
differential equations is one of them. Read through this tutorial and
see how the rate constant k in this tutorial is an example of a time
constant.
http://www.ugrad.math.ubc.ca/coursed...eqs/intro.html
--
John Popelish