In: Physics
The foundation of digital electronics is boolean logic. Using a binary number system (base 2), boolean logic can represent information processing in bits of 0 or 1, and those bits are manipulated using gates such as AND, OR, NOT, etc. (Could be replaced with: Every type of logic gate circuit can be made using only 2-input NAND gates and so every binary calculation can be solved by using them alone. Also, inverters and buffers exhaust all the possibilities for every single-input gate circuit.) which fully encompass all mathematical representations and computations (turing completeness). We can calculate binary logic in our heads, or use paper and pencil, but mechanizing it gives us speed and storage much larger than we can otherwise.
In the old days, such logic was mechanized in hardware using relays or vacuum tubes, each element representing a bit of 0 or 1 (true or false) depending on whether it is on or off. The problem was that they were physically large, required lots of power, and they tended to burn out quickly. Transistors revolutionized the way we represent these 0s and 1s in hardware because transistors are small, relatively low power, and durable semiconductors that can be manipulated to act either as conductors or insulators of electricity using a threshold voltage. If the voltage coming out of a transistor is above a certain threshold we label it 1/true/high, if it is below that threshold it is 0/false/low (we can reverse these labels if necessary and preserve the distinction).
The information revolution as we now know it that is reshaping our world. Transistors are the enabling innovation, and Silicon is the material substrate it is founded upon.