Basically, to "fine-tune" means to take something that is 98% correct and make it 100% correct. It comes from tuning an instrument: you need to tune it to get close to the correct note, and you fine-tune it to make it the exact note that will sound exactly the same as the other instruments playing the same note. In your example, the brain is already tuned, but language training improves (or "fine-tunes") it.

To fine-tune something means to make small adjustments in order to make something as perfect as it can be. When you fine- tune an instrument, you tweak it to get perfect pitch. When you fine-tune an essay, you polish it for greater clarity and readability. When you fine-tune a motor or engine, you calibrate it for optimum performance.