This is a point where they still didn't know what Doctor Who was supposed to be. Really, it's a point where they still didn't know what television was supposed to be--the BBC had been broadcasting since the medium had been invented, and certainly long before most people had a set to view their output, but they were still getting used to the idea that television was its own dramatic form and not simply an exotic way of bringing live theater to the British public. Sydney Newman, the new Head of Drama, was in the process of reconceptualizing the medium for a modern era, but the process very much occurred in fits and starts. And it showed in "The Edge of Destruction", which feels very much like an avant-garde play by Samuel Beckett brought to the screen.
It really shouldn't work--some people would say that it doesn't--but it is nonetheless fascinating to watch. Again, you have to credit the four regulars with whatever success the episode does have; they're all behaving out-of-character, with varying degrees of confidence in the script (William Russell, for one, seems very unclear on what exactly he should be doing with the part as written and settles for a sort of amiable giddiness) but their characters have been established so well over the previous eleven episode that you can tell perfectly that they're not themselves. It's an unsettling effect, watching Susan stare with cold-eyed suspicion at Barbara only a short while after the two of them relied on each other implicitly.
Most of the criticism of the episode--and let's face it, this is the point where we all admit that we're none of us watching the series in a vacuum; it's almost impossible not to engage with fan consensus, whether we agree with it or not--centers on the fact that the explanations offered for the events on-screen make no sense. Which is more or less true, for what it's worth. Susan and Barbara spend the whole first episode speculating that some sort of alien intelligence got into the TARDIS and is possessing them, moving from one body to another and turning them against each other. That's not what's happening, but it's frankly such a good idea that it's amazing nobody's ever tried it in the fifty years since. It's risky, coming up with a really interesting plotline and mentioning it in your story and then not using it like that. There's always the chance that the audience will wish you'd stuck with the red herring.
But even though nothing makes literal sense, it makes emotional sense in the same way that a Beckett play does. Something strange has happened to the TARDIS. Instead of being a refuge at the end of a long and horrific journey, it's now become a place of danger. Just what sort of danger is left inchoate and unspoken, but the very minds of the crew are affected. Nobody is certain of themselves or each other, and it seems like some alien presence is hovering over them. And what the crew doesn't know yet...is that they're right. There is an alien presence in the TARDIS. It is what's making them act giddy or paranoid or out-and-out homicidal. They just haven't figured out yet what it is. The whole thing hangs together as an aesthetic perfectly, even if a plot summary of it doesn't seem to follow linear logic. That's why in some ways, this episode feels strangely modern. It wouldn't be out of place next to "Ghost Light", or "Midnight" from the new series. Sometimes "sense" isn't about intellect. It's about emotions. And the emotions of "The Edge of Destruction" work perfectly.