The OED has a bit more info:
ME. a. OF. tregedie, tragedie (14th c. in Godef.), ad. L. tragœdia, a. Gr. τραγῳδία, app[arently]. goat-song, f. τράγος goat + ᾠδή ode, song.
As to the reason of the name many theories have been offered, some even disputing the connexion with ‘goat’. See L. H. Gray in Classical Quarterly VI. 60, and references there given.
The article is On the Etymology of Τραγῳδία, by Louis H. Gray, The Classical Quarterly, Vol. 6, No. 1 (Jan. 1912). (Not this Louis H. Gray, or he would have been 7 when he wrote it.) Gray considers a number of theories as to the connection with goats:
(1) The goat was a prize in early tragic contests... (2) there was a song of goats or goatmen... (3) there was a song of men dressed in goat-skins... the celebration being in honour of Dionysos Μελανάνγις; (4) there was a song of men dressed in goat-skins, such a costume being a survival of the archaic Greek dress... (5) a goat was led by the chorus to be sacrificed
Gray considers (4) to be the most plausible, assuming that τραγῳδία really does mean "goat-song". Another theory is that the word means "spelt song" from τράγος in its sense of "a mass of groats made of wheat" - however, this is probably a specialized meaning of τράγος "goat".
However, Gray prefers a different explanation, one that takes the etymological meaning of tragedy to be the antithesis of comedy. He looks for possible cognates of trago- in Old Norse þjarka "quarrel", þrekr "strength, courage, daring", Old English þracu "attack, pressure", and Old Irish trén "strong", from a Proto-Indo-European form *tereg-. In this theory, the original meaning of τραγῳδία was "the singing of bold (or terrible) things", and nothing to do with goats. I don't think this is accepted today; *tereg- isn't in any lists of PIE roots I can find.
ᾠδή is from ἀείδω "to sing" from *h₂wed- or *h₂weid- "to speak". Other -ody words include:
comedy, etymologically "revel song" (κῶμος "revel")
parody "wrong song" (παρά "beyond, amiss, wrong")
melody "tuneful song" (μέλος "limb, tune" from *mel- "limb")
rhapsody "sewn together song" (ῥάπτω (rhaptō, rhaps-) "sew together" from *wer- "to turn, bend").