It's not clear if your question is about how computers can do things behind the scenes (e.g, clock plus a set of instructions and some memory), OR, how computers bridge digital logic with analog circuits. I'll answer the latter.
The line between digital and analog circuits is actually really blurred. A computer can have a state, or some memory, that it can set to some value (true or false in the simplest case). So it may be simple to understand the following computer code.
if our calculation is finished, turn on that light.
- if (calculationFinished) lightOn = true
- this sets some memory (referenced by 'lightOn') to some value (true, possibly a 1)
But when you set some variable or some computer state to a value (such as true, or a number, etc..), it's still just a circuit state. Somewhere, that memory is stored inside a circuit. There's no special computerized ether outside the circuits - it's all analog under the hood.
In digital circuits, you typically represent true and false with some voltage like 0v (false) and 5v (true), to turn that analog world into digital. So if you have a value in software memory, somewhere there is actually a pin or a wire (or silicon or...) that either has 0v (against ground) or 5v (against ground) in our case.
In the simplest circuit you connect that pin to the LED and it turns on. Of course, that pin may be too high resistance (not enough current), so you feed it into a transistor as a signal that can let lots of current flow to power your light.
Computer World | Analog Circuit world
<amazing stuff>--some wire connected to lightOn -|- transistor -> LED
<here be dragons> | <here be other dragons>
They're in the same world.
If you're interested in how you can have memory in a circuit, e.g., that remembers that true even after the computer moves on, check out flip flops:
If you're wondering how a computer could possibly have dedicated wires for EACH BIT, when your computer can have giga BYTES of memory, it comes down to addressing and multiplexers: you set some memory to tell it /which/ memory you want to look at, and it acts like an old fashioned telephone operator to connect your wire to the right memory location. Check out multiplexers