Intuition for Church-Turing thesis for Turing machines
Imagine you are performing a computation by hand with a pencil and a stack of paper. [1] There is a limit on how many pieces of information you can keep in working memory at a time (sometimes claimed to be seven plus or minus two). So when you can't keep everything in your head, you write some of it down on a sheet of paper. And when you fill up a sheet, you put it in a pile for later reference and pull out another sheet. But there is a limit on how many sheets of paper you can look at at a time, too; you will have to flip between sheets as you work.
Turing machines are an abstraction of this idea of local computation. A Turing machine can write down as much auxiliary information as it wants, but it can only look at a finite amount of it at a time. A Turing machine head is like your brain's working memory—it can only store so much stuff before it has to write it down somewhere to avoid forgetting it.
The Church-Turing thesis says that any physically realizable computation does not require any &quot;essentially nonlocal&quot; operations. That is, any physically realizable computation can be broken down into a series of steps, each of which operates on $O(1)$ bits of information; there is no primitive operation that requires, say $O(n)$ arguments and cannot be reduced to operations with fewer arguments. [2] Or: anything you can compute in the real world can be computed given an unlimited stack of pencils and paper.

[1] Recall that the word &quot;computer&quot; in Turing's time referred to a human profession!
[2] A primitive operation that accepts an unbounded number of arguments is exactly what the oracle in an oracle Turing machine provides—hence why oracle machines can be more powerful than Turing machines.