How does a multimeter convert voltage/current to a numerical value?
Converting a analog signal to a digital number is, oddly enough, done by something called a analog to digital converter, or often A/D for short.

There are many ways this conversion can be accomplished, like successive approximation, sigma-delta, tracking, flash, and probably a few more basic strategies.  There are lots of variations and tradeoffs within the basic types too.  Most microcontrollers have successive approximation A/Ds built in.