In the spirt of ELI5, Digital computing, at its most fundamental, is about breaking complex tasks into a bunch of yes/no questions. For example, representing the concept of '4' digitally is akin to saying yes 4x [to the question 'one more?'] then saying no . . . This is found at the basic level where ASCII codes come from having a unique 'number' for every conceivable character, to unique values for every color in the palette, to every command you can imagine.

While we might in our brains have a larger concept of, say, 10,000, or pi or 'scroll left,' it's all a rapidly iterated and refreshed set of yes/no questions for the computer.

The computer keeps track of the values for these questions with a whole bunch of spots where there is no voltage for a no and a certain voltage for a yes. They're like on/off switches for your bedroom lights, only they are turned on and off with electrical signals instead of fingers.

You can imagine with tasks of any complexity or sophistication, the number of switches quickly become huge and cumbersome. As others have mentioned, they used to be relays or tubes that were nearly as large as actual light switches.

Then they discovered transistors that were made of semiconductors. And the properties of these semiconductors made it possible to simply use an electrical current. A tiny piece of the semiconductor can hold that charge value [ie, none or some, yes or no] until another current comes along and changes it.

Now all those switches can be replaced with tiny tiny little blobs of that semiconductor material specially arranged to serve as tiny tiny switches, and they can be packed really close together. Again, as others have said, billions and billions in the same space that used to be one switch or vacuum tube.

Here, the conceptual limits are being refined by our ability to fabricate. These switches, tiny as they are, still need to be structured in an orderly manner, so we've refined our ability to make smaller and smaller versions of the 'blob' that does the work of a transistor keeping track of the on/off yes/no values. You can't just 'make the tiniest transistors' and stick them on a motherboard, you have to make them with the right connections and arrange them so they keep useful track of those billions of values in a meaningful way to accomplish a goal.

So the work since the advent of the transistor has not so much been computing in a different way as it has been making the fabrication process smaller, more precise, and more efficient.