Learn Portuguese (Portugal): O capitão quer que o passageiro mostre o cartão de embarque. - The captain wants the passenger to show the boarding pass.

Because the sentence expresses a desire using quer (wants), and the person wanting the action (the captain) is different from the person doing the action (the passenger). In Portuguese, when you use verbs of desire like querer que (to want that) and switch subjects, the second verb must be in the present subjunctive. So, mostra becomes the subjunctive form mostre.

In English, we say "wants the passenger to show". Why doesn't the Portuguese sentence use the infinitive mostrar?

English uses the structure "want [someone] to do [something]". Portuguese doesn't do this when there are two different subjects. Instead, you must phrase it as "wants that [someone] does [something]" (quer que...). You only use the infinitive if the subject stays the same, for example: O capitão quer mostrar o cartão (The captain wants to show the pass).

How exactly is the present subjunctive form mostre created from mostrar?

For regular -ar verbs like mostrar, the present subjunctive is formed by swapping the usual a vowels in the endings for e vowels. For a "he/she/it" subject like o passageiro, you take the "I" form of the normal present tense (mostro), drop the -o, and add -e, which gives you mostre.