It's called post-training, typically through some form of reinforcement learning, and is a significant part of modern LLM development.
You have the first stage, pre-training, which is learning from next token prediction. That's where the model memorises a lot of facts about things and generally gets good at forms of writing. It's like reading a lot of books on programming and reading through a lot of source code. It's learning how to autocomplete code, essentially. Doing that requires a developing a reasonable understanding of code, but it's also learning how to autocomplete bad code as well as good, and won't make it a "good" programmer.
Pre-training uses a method called Cross-Entropy Loss to update the weights of the network.
Then comes post-training. This is where the model is trained against huge sets of example problems, like fixing a bug, adding a new feature based on a spec, etc. They are set the task and try to complete it inside a training environment. Once they're done, their complete solution is evaluated (either by humans, or by some separate evaluation model that was developed based on human feedback) and they are updated based on whether the solution was good or not.
Post-training uses a different method called Proximal policy optimization to update the weights of the network.
So these really are very different forms of learning, and mainstream LLMs are not post-trained to be good at chess. They could be. You could easily create a reinforcement learning environment that evaluated and improved their ability to play and win at chess. The result would be a very strong chess playing AI, something we know is possible because the strongest chess playing programs we have are neural network based, but it is not a priority for AI companies.