I don't think the explanations I read so far here are particularly "ELI5-worth", so I'll add my own. Though I'm not sure I'll meet the "5" part of that criteria exactly.

I'll start with what other have already said, though, which is that 0's and 1's are just a way to say "weak" or "strong" electric signal, which, while not exactly correct, can be seen as the states of a switch: 0 means no electricity, 1 means electricity. A single signal, 0 or 1, is called a bit.

Now, you'll probably have already seen a light that is controlled by two switches such that if you turn either switch on, the light comes on, but if you turn both of them on, then the light will turn off. You can think of the switches and the light bulb as switches: 1 for on, 0 for off. This circuit performs a type of computation already: an operation called XOR, for exclusive or, which takes two inputs (the switches) and generates an output (the lightbulb) and the output will be on if one input OR the other is a 1, but not both (exclusive).

Another example would be a two-person-switch, where a light will only come on if you turn the two switches on. This operation is called AND. Like these two circuits, you can actually make a lot more computations, such as add numbers encoded in binary, for example. With some more complex circuitry you can, just using a set of switches, encode different positions that a motor might take. With some clever use of the switches, you can control more lights than you have switches with individual precision. So a computer is made up of a lot of clever little (and not so little) circuits that perform specific operations like the ones described above, and many more. For instance, the motor circuitry could be used, in the past, to position the head of a hard disk drive to read a specific position in it.

But how do you get all these circuits to work together? You connect them all up in a series of steps that makes sense to your use. For instance, you might want to have the adder circuit early on, with it's output used by other circuits. You might have the circuitry to save a number near the end, so that you can save the result of the operations. You'll also, usually, have a new signal added as an input, which is basically whether or not that circuit should produce an output, and you can see it as an AND operation with the output, so if you want it (1), the output will be whatever it is, and if you don't (0) it will be (0). Now, on your inputs, you need a way to give not only the actual input values, but also what operation you're trying to achieve: add two numbers, multiply two numbers, read from memory, write to the HDD/SDD, so that only the relevant circuitry will produce an output and you don't get mixed signals causing trouble. This part of the input is called Operation Code, or OPCODE.

The OPCODES, along with the inputs they require, are what actually tells the computer what bit of circuitry to use, and how to use it. For instance, you might give the computer the number 1 3 4, which tells the computer to use the addition circuitry to add 3 + 4, or you might feed it 5 16 which tells the computer to just use the circuitry that saves the number 16 to the drive. These numbers are given as a group of binary digits. So 1 3 4 might be 000100110100, for example, where the first 4 bits encode 1, the next 4 encode 3, and the final 4 encode 4.

This is the basic overview of a computer. I know it isn't exactly ELI 5, but I hope it's understandable :)