Translation:He showed us how to ride a horse.
- this is a compound tense built with verb "avoir"
- therefore, the following rule applies: the past participle is invariable EXCEPT when the direct object is placed BEFORE the verb.
- the direct object of "a montré is "comment monter à cheval" (question: showed what? how to...).it is placed after the verb and it is neutral (a whole sentence = equivalent of "it")
- "nous" is an indirect object (question: showed to whom? to us)
Here are a couple links that may help.
As a native English speaker "showed" is in the past tense, but "shown" is a past participle. I know for sure that in the United States (I don't know about anywhere else) we would NEVER say "has showed" but we would say "has shown." We only use "showed" if we were to say "he showed me his car."