Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer.
Here is a short documentary on Conway’s Game of Life, beautifully explaining how the logic gates of the Turing machine were implemented to run inside GoL.
For more in-depth explanations, read: Rendell, P., 2014. Turing machine universality of the game of life (Doctoral dissertation, University of the West of England).