Meanwhile, in the TARDIS: Part 2

The Doctor explains that part of the reason he takes on companions is to live vicariously through their amazement of time and space. He explains that he's lost the ability to see the wondrous aspects of life due to living several centuries.

This is something very real for adults in real life. It becomes harder to maintain optimism and a sense of joy in later years as you experienced more and more. So when you get to show or teach someone else about what you know, it gives back some of the joy you lost.