Here is Penrose argument in the first few minutes of this video summarized to the best of my ability/understanding:

1. Godel’s Incompleteness / Halting problem etc, shows there are truths which cannot be mechanically derived using an algorithm or a finite set of axioms. e.g. claims like all even numbers greater than 4 can be written as a sum of two prime numbers (a.k.a Goldbach’s conjecture).
2. All of the truths in Physics can be mechanically derived using an algorithm. e.g. future/past/present state of a quantum mechanical/relativistic/classical system.
3. Let us assume our minds use Physics
4. But our minds can derive truths which cannot be mechanically derived using an algorithm or a finite set of axioms. e.g. claims similar to Goldbach’s conjecture which have already been proven.
5. Therefore whatever process our minds are using to prove such things, and derive such knowledge must not be using Physics, because if it were using Physics then such a mental process could be mechanically derived using an algorithm, because Physics can be mechanically derived using an algorithm. Therefore our assumption 3, is wrong.