Learn Polish: Urzędnik wydał mi zaświadczenie. - The clerk issued me a certificate.

mi is the short, unstressed dative form of "to me". We use short forms when the pronoun is in the middle of a sentence and isn't being emphasized. You would only use the long form mnie at the beginning of a sentence, after a preposition, or for strong emphasis (like saying "The clerk issued it to ME, not to you").

I recognize the verb dać (to give) inside wydał. How does the prefix wy- change the meaning here?

The prefix wy- generally adds the meaning of "out" or "outwards". So wydać literally means "to give out". In an administrative context like this one, it means "to issue" a document. (Note that wydać can also mean "to spend" when talking about money!)

What case is zaświadczenie in, and why didn't its ending change?

It is in the accusative case because it is the direct object of the sentence (it is the thing being issued). However, zaświadczenie is a neuter noun. For all neuter nouns, the accusative form looks exactly the same as the nominative (dictionary) form, which is why the ending doesn't change here.

zaświadczenie is quite a mouthful! Is there an easy way to remember this word?