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The paper A universal cellular automaton in the hyperbolic plane says:

Our simulation consists in simulating the execution of a register machine. It is known that two registers are enough to simulate a Turing machine, see [10], and so, by such a simulation, we shall obtain a universal computation.

It links to [10] which is Computation: Finite and Infinite Machines, a $200 book from Minsky with no reasonable/free version seemingly available on the web, so no easy access.

How are two registers enough to simulate a Turing machine? How do we know that? What is the sketch laid out in the Minsky book that explains how this works?

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The reduction is sketched on Wikipedia.

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