Learn Russian: Вчера она мне это сказала.

мне is the Dative case form of я (I). In Russian, verbs of communicating, giving, or showing use the Dative case for the person receiving the action. You use мне because she said the words to me.

Does Russian have a word for "to" in this sentence, like "said to me"?

No extra word is needed! The Dative pronoun мне already has the built-in meaning of "to me" when used with verbs like сказала.

Why are the words мне это placed before the verb? Can I put them after?

Russian word order is very flexible. Short pronouns often naturally group together right before the verb. It is also perfectly fine to say Вчера она сказала это мне, which just puts a tiny bit more emphasis on the fact that it was said specifically to you.

Why does the verb сказала end in -ла?

This is a quick reminder about how the past tense works! Past tense verbs change their endings to match the gender of the subject. Since the subject doing the action is она (she), the verb takes the feminine past tense ending -ла.