Learn French: Le pharmacien m'explique l'ordonnance. - The pharmacist is explaining the prescription to me.

In French, the pronoun me (which becomes m' before a vowel) already means "to me" in this sentence. The verb is expliquer à (to explain to). When you use an indirect object pronoun like me, the "to" is built right in, so you don't need to add à.

Why does m' come before the verb instead of after?

In French, object pronouns like me, te, nous, and vous go directly before the verb they are connected to. While English says "explains to me," French phrasing is always "m'explique" (literally "to me explains").

Is m' short for me or moi?

It is short for me. Whenever me (or te) is placed right before a verb that starts with a vowel or a silent h, it drops the "e" and uses an apostrophe to make pronunciation smoother.

Does ordonnance mean an order or a rule?

It can in other contexts, but at the pharmacy or doctor's office, une ordonnance is the standard word for a medical prescription. It comes from the idea that the doctor is "ordering" a specific treatment for you.