Translation:The teacher reads a newspaper to the girl.
It's part of the verb: "vorlesen" (to read to sb., to read out to sb, to read out loud to sb.) is a separable verb. "Lesen" just means "to read", but "vorlesen" implies that you're reading to somebody else.
For more information on separable verbs, see: http://www.deutschseite.de/grammatik/trennbare_verben/trennbare_verben.html
Because you change the article based on case. Here, the girls are the indirect object of the verb "to read," so you have to use the dative case. The dative case of "das" is "dem."
Similarly, "die Zeitung" needs to replace "die" with the German indefinite article (ein). But you have to make "ein" into feminine. That makes it "eine Zeitung." Then you need to make it the accusative case. Fortunately, the accusative case of "eine" (feminine) is "eine." So "eine Zeitung."
Not without changing the meaning of the sentence. With separable verbs (see the second comment from the top), the preposition that separates from the verb is put at the end of the sentence. If you put the "vor" in the spot where your sentence has it, the meaning becomes "The teacher reads a newspaper in front of the girl."
In this sentence Mädchen is dative;"The" as neuter dative is "dem" and as plural dative "den".
Notice that for each case the plural articles are always different from the Masculine and Neuter articles. Which is handy because many masculine and neuter plural nouns are the same in singular and plural. Feminine nouns almost always change in the plural which is handy because The feminine articles are usually the same as the plural articles. I' sure that is not a coincidence.
Like all nouns ending in the suffix -chen, "Mädchen" is neuter.
The forms of the definite article for neuter nouns are:
nominative: das (for the subject, e.g. The girl is interesting. And after the verbs "to be" (sein) and "to become" (werden), e.g. This is the girl)
accusative: das (for most direct objects, e.g. I see the girl)
dative: dem (for indirect objects, e.g. I write a letter to the girl)
genitive: des (to indicate possessions, e.g. This is the girl's book)
PLURAL (identical for masculine, feminine and neuter nouns)
nominative: die (usage: see above)
accusative: die (usage: see above)
dative: den (usage: see above)
genitive: der (usage: see above)
As you can see, the only case that fits here is the dative singular: "Der Lehrer liest dem Mädchen eine Zeitung vor." This makes sense, because we're dealing with an indirect object here: The teacher reads a newspaper to the girl.
(Note that the above is a simplified and not a comprehensive description of how the cases are used).
See also this table: