The Halo array wipes out all life in the galaxy when activated. This is because the Flood pretty much destroyed the Forerunners long ago, and it was a last-ditch effort: they can't kill the flood, but they can kill the food, and starve the flood to death.

Anyway, the Covenant misinterpret Forerunner stuff they found and see mentions of a 'Great Journey" they would go on if they activate the array. They believe that walking the path/ going on the great journey would allow them to follow in the Forerunner's footsteps and become gods. Because they think the Halos allow them to transcend mortality.

Thing is, Forerunners liked humans. At first contact, a Covenant luminary (A device that detected forerunner tech) detected millions of objects on the world we would come to know as Harvest. It turns out that the luminary was detecting the humans themselves, but the Brutes (You'll meet them further down the line if you play Halo 2) decided the humans were hiding the relics. Now. On the Covenant's capital, there is a Forerunner ship docked- it houses a Forerunner AI named Mendicant Bias, that was tasked with the Forerunner militaries before the array fired. This AI basically told three of the Covenant's leadership "Hey. Everything you think is wrong, and te humans are the cool guys. Oh, and fuck you." Naturally, the Prophets know that if this knowledge spread, it would result in the dissolution of the empire (I'll get to that) and so they decide to keep it quiet and encourage the ideal tht humans are heretics and must be wiped out. So at first, you're simply fighting for your own continued survival. Later, you're fighting to make sure you don't wipe out every creature in the galaxy.

*The two main warrior castes - Sangheili and Jiralhanae (Elites and Brutes) and the two 'cannon fodder' castes - Kig-Yar and Unggoy (Jackals an Grunts) despise each other. Without a common goal, the Covenant would collapse in a bloody civil war.