I don't think "he shows himself" could possibly be right here. Also, like "he shows him," it has more than one possible meaning. Do you mean "he shows himself (something)"? I think that would be "il lui-même montre (quelque chose)." If you mean "he shows himself (to someone)" then I'm less certain, but perhaps "il se montre (à quelqu'un)."
Le is the direct object form of him/it . Lui is the indirect object form of him.
He shows him would mean that he (the subject) was showing a male something (the object) maybe a person, animal etc to someone else. Police will bring a perpetrator out to show to the media. When that happens he (the prosecutor) is showing him (the perpetrator) to the public. Il le montre (he shows him or it)
In English, he shows him could also mean he shows to him. The listener/reader has to figure out which meaning is intended.
But in French it is clear. Il lui montre where lui is the indirect object form. (lui = to him in this sentence) Il le montre (le is the direct object form shows him to someone else in this sentence)
Because le is a pronoun standing in for a noun, we don't know whether the noun it represents would be registered as him or it in English. He shows the fish/it to the class, he shows the winner/him to the audience. Either one is le.
In sentences of three or four words with no context it can be hard to figure out which one it is. In conversation you will almost always have enough context to know right away what the noun is and thus how to treat it.