Intuition for Church-Turing thesis for Turing machines

I can very clearly see "why" mu-recursion is a universal model of computation, i.e. why the Church-Turing thesis — that any physically computable algorithm can be executed with mu-recursion — holds for mu-recursion. It reflects exactly the type of algorithms that I can carry out with my own brain.

I cannot see an analogous intuition for understanding why the Turing machine can execute any physically computable algorithm — i.e. how did Turing "see" that the Turing machine was a good definition to use? Is there a good way to "imagine" the algorithms I perform in terms of the Turing machine, as opposed to general recursion as I am used to?