Does the "use it or loose it" principle apply to the brain? Does not using the brain decreases mental functioning? If so, is it recommended to keep using the brain, for example by learning new things or engaging in challenging activities (puzzels, quizzes etc.) up until old age?
Yes, brain power is eroded via a lack of practice. This occurs through the processes called synaptic pruning and brain plasticity.

I will leave you with a very basic answer as I am unsure of your level of understanding of cognitive processes. First, you must understand neurons. Then, you can begin to understand synaptic pruning and brain plasticity. Finally, here is a link to a bunch more information and brain plasticity exercises.