In order to combat the phalanx's famous right-hand drift, Greek commanders traditionally placed their most experienced, highly regarded and, generally, deadliest troops on the right wing, as this was the place of honor. By contrast, the shakiest and/or least influential troops were often placed on the left wing.

In a major break with tradition, Epaminondas massed his cavalry and a fifty-deep column of Theban infantry on his left wing,[2] and sent forward this body against the Spartan right.

Basically the Thebans deliberately deployed their strongest forces against the spartan strongest forces. This was probably not only for tactical reasons but for morale reasons. They wanted to show to the rest of their greek allies, that they are willing to face spartans face to face, instead of sending some other random(and weaker) greek phalanx against them.

It is important to note that the only reason Sparta became the biggest player in Greece, was that Athens(the popular kid, ie the US) had managed to piss off and screw over everyone else in Greece. Everyone still hated Sparta but they hated Athens even more. So they allied with Sparta in order to destroy Athens.

Meanwhile Persia saw the opportunity to stir shit up and supported Sparta. Athens was after all the reason/excuse that Persia invaded Greece(twice, both failed). Because Athens had decided to help its colonies revolt in Anatolia, which were under persian control at that time.