I read the paper and so can provide a layman’s summary.
The authors considered game states where there are no more decisions left to be made by either player, and no more randomness. There is only a sequence of forced actions and triggered effects left to be resolved.
The authors say: assume for sake of contradiction that there is an algorithm that can take such a game state as input and output who will win the game. They prove that if such an algorithm exists, then the algorithm can be used to solve the halting problem. But the halting problem is provably undecidable (no algorithm can solve it). This implies that the assumption of the existence of an algorithm that can determine the winner from a game state is false.
The sketch of the proof is that they cleverly construct a game state that can represent the internal state of a Turing machine, with the mechanics of the remaining triggered effects mapping to the execution of the Turing machine’s instructions. A Turing machine is basically a representation of any algorithm (or computer program). Here is a snippet of their construction:
Each Rotlung Reanimator needs to trigger from a different state being read – that is, a different creature type dying – and needs to encode a different result. Fortunately, Magic includes cards that can be used to edit the text of other cards. The card Artificial Evolution is uniquely powerful for our purposes, as it reads “Change the text of target spell or permanent by replacing all instances of one creature type with another. The new creature type can’t be Wall. (This effect lasts indefinitely.)” So we create a large number of copies of Rotlung Reanimator and edit each one. A similar card Glamerdye can be used to modify the colour words within card text.
Thus, we edit a Rotlung Reanimator by casting two copies of Artificial Evolution replacing ‘Cleric’ with ‘Aetherborn’ and ‘Zombie’ with ‘Sliver’ and one copy of Glamerdye to replace ‘black’ with ‘white’, so that this Rotlung Reanimator now reads “Whenever Rotlung Reanimator or another Aetherborn dies, create a 2/2 white Sliver creature token”4. This Rotlung Reanimator now encodes the first rule of the q1 program of the (2, 18) UTM: “When reading symbol 1 in state A, write symbol 18 and move left.” The Aetherborn creature token represents symbol 1, the Sliver creature token represents symbol 18, and the fact that the token is white leads to processing that will cause the head to move left.
You can sections III and IV of the paper for full details.