This is a weird one but a good one. Matt Dillon is out of town for a couple of days when a mysterious man comes into town and comes right up to Doc as he is dining and says he intends to kill him. Apparently the man thinks Doc is actually the judge who sentenced him to prison, even though Doc tells the man he is the town doctor and that he can ask anybody in town who he is. The man is not persuaded in the least and says that he knows what he knows. This is why I don't necessarily trust eyewitnesses, but I digress.
Doc blows all of this off because the man does not have a gun, at a time when just about everybody wore one. How did the stranger intend to kill him if he didn't have a gun? It turns out the man has escaped from prison recently and therefore does not have a gun - yet. Fortunately for Doc, Chester is sitting next to Doc when the man threatens him. Chester is alarmed by this entire situation and keeps tabs on the man. When he sees him quick draw during a confrontation at the Long Branch, Chester realizes this fellow is a real threat to Doc, so he swings into action with a plan that could put Doc in harm's way, but it definitely puts Chester very much in danger as well.
Now we'd expect this of Matt - He is always calm and cool headed under pressure. And also, he has taken an oath to uphold the law. You have to remember, Chester is not an actual lawman. He can't make arrests. He never took an oath. He was simply hired by Marshal Dillon to help out, although for sure he is a very trusted employee and good friend. So Chester is doing this partly out of his loyalty to Dillon, but also there is a large chunk of friendship between himself and Doc, for all of the grousing they do with each other.
In the end there is much unsaid in this episode, at least not in words. And that is part of what makes it so excellent.