The reasons depend on the story you are reading.
The medieval tales give no reason. In the early tales Mordred is made regent of Britain when Arthur goes overseas to fight the Romans, and later is informed that his nephew Mordred has usurped the throne, taken Arthur’s queen Guenevere as his wife, and made an allegiance with the Saxons.
No reasons are given for this.
Mordred seldom appears in the verse Arthurian romances. But where he does appears, as in “Claris and Laris”, he is a cruel and vicious knight. In “Claris and Laris” he is a would-be rapist.
In the “Prose Lancelot” and later prose romances, Mordred is nephew to Arthur, but at the same time Arthur's illegitimate son, begotten by Arthur unknowingly on his half-sister, King Lot’s wife. Mordred is informed of this for the first time by a hermit, whom he immediately kills. Mordred is generally pictured as a vicious and cruel knight, a killer and a rapist. One wonders why Arthur would be so stupid as to put such a person in charge of his kingdom. One also wonders why so many continued to go along with Mordred, after they learned that Arthur was still alive. The only answer provided is that Mordred had given out great gifts, almost emptying Arthur’s hoards of treasure, and that there was much good in Mordred, though we don’t see it.
In Boece's Scottish history, Gawain and Mordred are cousins to Arthur rather than nephews, and Mordred may be the elder. Their father King Lot had fought against Arthur early in Arthur's reign and had made peace on the terms that if Arthur had no children, then Lot or a son of Lot would be heirs to the kingdom. When Arthur, late in his reign, names Constantine as his heir, Mordred is enraged and that is the motive for his rebellion. In this version Mordred can be said to have right on his side.
Modern novelists tend to delve into this more deeply than the medieval tales, and sometimes show far more sympathy towards Mordred. Sometimes the main reason for the final battle is just a difference of opinion between Arthur and Mordred, Mordred having taken over the kingdom honestly believing that Arthur was dead. Sometimes Mordred rebels because he knows that he is Arthur's nearest living kin, and therefore thinks he should be heir. In some tales the cause is dated back to Uther Pendragon wrongly begetting Arthur on the wife of Gorlois, Duke of Cornwall. The descendants of Gorlois tend to be hostile toward Arthur, and Mordred is portrayed as taking up the cause.