Learn Italian: Il batterista ci ha mostrato come tenere il ritmo. - The drummer showed us how to keep the rhythm.

This is an implicit indirect question. Because the people receiving the instruction (ci, meaning "to us") are the ones who are meant to perform the action, Italian uses the question word come followed directly by the infinitive tenere. It translates perfectly to the English structure: "how to keep".

If the sentence means "how he (the drummer) keeps the rhythm", could we still use come tenere?

No. In implicit indirect questions following verbs like mostrare (to show) or insegnare (to teach), the infinitive usually refers to the indirect object (how we should do it) or has an impersonal meaning ("how one keeps"). If the drummer was just showing us how he himself keeps the rhythm, you would need an explicit clause with a conjugated verb: ci ha mostrato come tiene il ritmo.

Why doesn't the past participle mostrato agree with the pronoun ci?

The pronoun ci is an indirect object here (the drummer showed to us). When using the auxiliary verb avere, past participles only change their endings to agree with direct object pronouns (like lo, la, li, le). Since ci answers "to whom?", mostrato stays in its default masculine singular form.

Why does the word for drummer end in if the article is masculine ()?