This explanation may not be too academical but this is how i understand it (didn't go to university either).
The first step involves Boole's algebra that proved in the 19'th century that any mathematical operation can be expressed using a series of symbols and some associated operators. Thus the base 10 arithmetic with the common +, -, *, / operators can be expressed in using just two symbols (0, 1/ true, false) and logic operators (AND, OR etc) leading to the Boolean logic that is the mathematical foundation of digital computing.
The second step is Alan Turing's work that constructed a mathematical model of an imaginary machine that could process symbols. The naive description of a Turing machine would be any automaton that has memory (to store it's state), operates on a series of symbols that represent it's state and interprets another series of symbols that define the transition from one state to another.
The modern digital processor is an implementation of a Turing machine that uses Boole's logical algebra as the symbol and operator base (take a look at a How Stuff Works detailed explanation of Boole's logic at work). The reason for this is that the true/false logical states can be easily mapped to electrical signals (+ , -) and their associated operators can be implemented in transistor circuits that given a binary input (electrical signal) can output the result according to the boolean operators. So any modern computer is a machine that has memory
to store it's state (encoded in binary electrical/magnetic form) and an array of circuits that accept electrical impulses and operators (the processor instructions) and output results accordingly at a very fast speed.
Thus every program written in any computer language is eventually translated by the compiler or runtime into a series of boolean operations for the processor to execute