I believe it was something to do with fame, there are two POVs, Fergie and Beckham, After Beckham married his wife and started to build a brand he became super famous, SAF thought Beckham thought he was too big for the club and that he had to go.

The contract Beckham had took a long time to get through a lot of that was to due with Beckham's image rights and I believe he became the highest paid player in the world at the time, he got injured during his last season and lost his place in the team to Soleskjar, he had a few arguments with Fergie and at one point Fergie stormed into the dressing room after a defeat and kicked a boot that hit Beckham in the face.

Fergie thought Beckham believed he was too big for the club, Beckham maintained that it wasn't the case as I understand the story anyways.

EDIT: This thread gives a quality explanation from both points of view: https://www.reddit.com/r/reddevils/comments/4jtt4q/in_their_own_words_the_breakdown_between_david/?