Turing's goal here is to justify why a Turing machine captures the "intuitive notion of a computation". To his audience, a "computer" is a person with a stack of paper and a pen, following an arbitrary procedure. Turing shows here that, even though a person can potentially draw infinitely many drawings on a sheet of paper, that doesn't give them any more computational power then writing symbols from a finite alphabet into a grid of squares.

The machine he introduces also writes symbols onto paper with a pen and reads them. So he really is talking about pens and sheets of paper, it's not an analogy for something else.