When a show touches your heart it's most certainly because you relate to it. There is an understanding between the show, it's creators, and yourself. All the best shows are allegories of life. Take Buffy the Vampire Slayer for example, the show is an allegory for life as a teenager. Life as a teen can be hell, so where did Buffy live? The Hellmouth. Aside from the very well known Joss Whedon, Russell T. Davies is my other favorite allegorist writer.
What RTD did for Doctor Who is something that I think will live in the hearts and minds of Whovians forever. He gave us his heart(s) and road up and down the roller coaster of life with us. Everything from family affairs to tumultuous friendships to unrequited love. Loosing and gaining friends, family, and love all along the way. What I've realized recently is that the changing of the guard of companions and regenerations symbolize life's chapters.
When you close one chapter you're letting go a part of yourself and others. Then at the opening of a chapter you gain a new piece, all the while changing. It's the very definition of life and growing up. The Doctor isn't just a Time Lord, he's a physical representation of the metamorphism of life. It's sometimes hard to define the moment of change, but what is certain is that change does happen (though we don't usually get a new face when it happens). Letting go of the old chapter maybe hard but it's inevitable and all a part of progression (regeneration).
Life also takes you where you need to be, not always where you want to be, just like the TARDIS. How many times can you think of where you hoped and planned to be at some milestone of life in 6 months or a year, only to be derailed by the universe. It can whisk you off without any notice at all! In other words, the TARDIS drives the world, not us.
Life is such a strange being, it has no rhyme or reason, it just is. A big blue box of it. And that's why I adore Doctor Who, because it shows us that it's all okay. Just keep crossing new paths, meeting new people, and growing as a person. It doesn't have to make sense all the time, it just has to keep moving forward, in time and space.