Here "train" is the act of learning and practicing. The word "to" is used to mean "until". Furthermore, "exhaustion" is a stated of being (a noun).
Or:
We had practiced [some skill] until we became exhausted every day for many years to be ready for game.
This means that on each day, for many years, a group of people trained so much that they were exhausted. So each night for a year they would have been very tired.
However:
We had trained to exhaust every day for many years to be ready for game.
Exhaust is a verb, "to exhaust" is the infinitive form. This sentence means that the action they are training to perform is "to exhaust" - essentially they are learning how to make other people exhausted.
We had trained for exhaustion every day for many years to be ready for game.
Using for instead of to also changes the meaning, "to" means that exhaustion is the result of the training, "for" means that exhaustion is the goal of the reason for the training. With "to" the sentence implies that they are training for the game.
Essentially this means that the group is learning how to handle being exhausted (increasing their stamina) so that they will perform better in the game. While the original version means that because they were training for the game, they became exhausted.