This is a strange little pair of episodes. I wasn't very impressed when I first saw them, because frankly I had no clue what was going on. Even now that I understand it, the story still feels like improvisation, where David Whitaker just started writing strange dialogue for the characters and ran with it without knowing where he would end up. All of this is not to say that the story doesn't have merit, because it does, and I certainly like it now more than I did then. As a character piece it's fairly strong, however it's still not a very satisfying story.
I suppose the main purposes of the story are to flesh out the characters and to show us more about the TARDIS, and it manages to do both. With no cavemen or Daleks to steal the show, it's left to the four regulars to carry both episodes on a few small sets, and they manage this fairly well. The normally safe haven of the TARDIS has become a dark and uncertain environment, which is in itself an unusual event. We have four good actors playing four great characters in a mysterious situation, which means it ought to be exciting, but it somehow never really is. On the other hand, it is interesting to a certain degree, mainly due to the strange behavior of Susan and Ian. Susan's convulsive stabbing of the couch with a pair of scissors is pretty disturbing.
A quick examination of the plot is warranted here. The ship leaves Skaro, headed back to Earth. Along the way it crashes, knocking everyone unconscious. This evidently causes some strange side effects, such as pain, or temporary amnesia. Some time is spent trying to figure out what has happened, and the suggestion is put forth that something has gotten inside the ship, possibly hiding in one of the passengers. What has actually happened is that the fast return switch has stuck, and the TARDIS is trying to prevent its own destruction, hence the odd behavior it causes Ian to engage in, or the melted clock it produces. In the end, after a trying experience, the crew are closer together and the Doctor is less hostile and more open than he was.
It's sound enough I suppose, but the idea of an intruder being aboard is never conveyed very well, and the fast return switch explanation doesn't really hold up to scrutiny. If it's stuck, it's malfunctioned, and should have registered on the fault locator. It's an attempt at a clever ending that doesn't quite hold up.
Where the story does hold up well is in character development. By bringing the Doctor and Barbara to a catharsis of sorts, and by putting the Doctor clearly in the wrong, and by having him realize it, the crew is finally able to have it out and settle their differences, and emerge from the experience as friends rather than reluctant travelling companions. The Doctor eats a little humble pie and becomes much more accommodating to his travelling companions, leading the way later to friendship and camaraderie rather than antagonism.
I think "The Edge of Destruction" has more value in context of the season as a whole rather than as a story in its own right. It holds up on its own, and I'm glad it still exists, but it works far better as a bridge between "The Daleks" and "Marco Polo", allowing our characters to resolve some differences and form friendships before moving on to future adventures. While it's a decent little story, it falls short of the surrounding episodes. 7 out of 10.