There are a few different possible themes to this (outside of the simple suspense of finding out what happens to the narrator).
First, you can argue that a theme is that little decisions made out of foolishness and overconfidence can (but don't always) have a huge price. The narrator admits this at one point in the story. As we read the story, we know that he has made a mistake. He should have just let Perkins drive. But we make mistakes like that all the time and don't get badly punished. In a way, then, this is a story about the vagaries of luck and fate. Two actions on our part, both equally stupid, could have very different effects due only to luck.
Another possible theme has to do with the nature of courage and duty. In this view, Perkins is as important as the narrator in showing the theme. Perkins, for example, offers to try to drive the car while his master attempts a jump to safety. The narrator declines and offers Perkins the same chance. He too declines. They seem to feel a sense of duty to one another and they are both (one can argue) showing bravery in accepting that duty. Perkins shows his feelings towards the end when his first words to the rescuers are a question about how his master is.
Either of these could be seen as the major theme of the story.
Sir Arthur Conan Doyle was a convinced believer in spiritualism, which is based on the belief that human souls live on after death and that living persons can communicate with them, usually through "mediums." It is often overlooked that Doyle begins this story with the following brief explanation:
She was a writing medium. This what she wrote:--
Then the story immediately becomes a first-person narrative by a wealthy country gentleman who loses control of his car and after a harrowing drive down a winding road is killed in a smash-up in front of his home. The narrator is already dead when he is telling this story through the medium, but the reader does not fully realize this until the end, when the narrator meets his old friend Stanley who died during the Boer War.
"Stanley!" I cried, and the words seemed to choke my throat--"Stanley, you are dead."
He looked at me with the same old gentle, wistful smile.
"So are you," he answered.
The excitement of the wild ride down the hill makes the reader forget that the story is being narrated through a so-called "writing medium," so it will come as a shock to realize that the narrator is fully conscious but has joined the invisible ranks of the dead.
Doyle sincerely believed in life after death and in the possibility of communicating with the dead through spiritualist mediums. He lost his beloved son during World War I, and this experience had a lasting effect on his mind. The main theme of his short story "How It Happened" therefore would appear to be that human souls survive after death.