How is the word tragedy connected to goats and songs?

The answer is that tragedy comes ultimately from the Ancient Greek word τραγῳδία ‎(tragōidía – epic play, tragedy) which comes from τράγος ‎(trágos – male goat) and ᾠδή ‎(ōidḗ, – song).

Apparently the goat reference comes from satyrc drama, which featured actors dressed in goatskins playing satyrs. Or because at Athenian festivals a goat was given as a prize for the best play or performance, and then sacrificed, and a τραγῳδία was a lament for the goat.

Another idea from Athenaeus of Naucratis (2nd–3rd century BC) is that the original form of the word was trygodia from trygos (grape harvest) and ode (song), as the festivals took place during the grape harvest

Tragedy entered English during the 14th century as tragedie, when it meant ‘a play or other serious literary work with an unhappy ending’. It came from the Old French tragedie, from the Latin tragoedia ([theatrical] tragedy). It came to mean an unhappy event, calamity or disaster at the beginning of the 16th century.

Sources: Wiktionary, Online Etymology Dictionary, Oxford Dictionaries, English Word Information