Learn Spanish (Latin America): Ellos desean que la visita compre la cena.

Because this sentence uses the present subjunctive! The first part of the sentence (Ellos desean que...) expresses a want or desire, which forces the following verb into the subjunctive mood. For regular -ar verbs like comprar, the normal -a ending changes to an -e. Since la visita is the one doing the buying (acting as he/she/it), the verb becomes compre.

Why is there a que in this sentence? Doesn't the English translation just say "to buy"?

In English, we say "They want the guest TO buy dinner." But in Spanish, when one person wants a different person to do something, you can't use the infinitive form of the verb. You have to connect the two parts with que (that). Literally, the structure is "They desire THAT the guest buys dinner."

Doesn't la visita mean "the visit"? Why does the translation say "the guest"?

Yes, la visita literally means "the visit", but it is very commonly used in Spanish to mean "the guest", "the visitor", or "company" (as in "we have company over"). It is always a feminine word (la visita), even if your actual guest is a man.

Could I use quieren instead of desean here?