Learn Hardcore Croatian: Danas me bole leđa.

Because leđa (back) is a plural-only noun (pluralia tantum) in Croatian. It’s treated as neuter plural, so the verb agrees in 3rd person plural: bole.
Compare: Boli me zub (tooth, singular), but Bole me zubi (teeth, plural) and Bole me leđa (back, plural-only).

What case is me, and why is it used here?

me is the unstressed accusative form of ja (I). With boljeti, the body part is the grammatical subject, and the experiencer (the person who feels pain) is in the accusative: Boli me glava, Bole me leđa.

Why not mi instead of me?

mi is the dative clitic (“to me”). Standard Croatian uses the accusative (me/te/ga…) with boljeti, so say Boli me glava, not Boli mi glava. You will, however, use dative with states like weather/temperature: Hladno mi je, Meni je vruće. Some speakers use dative with boljeti colloquially, but avoid it in standard Croatian.

Where does the clitic me go? Is the word order fixed?