Why did Ringo leave in 1969?

Abbey Road was not a happy working environment during the recording sessions for The White Album. Ringo, the least involved in the squabbling and backbiting, suffered the most from the emotional fall out:

I couldn't take it any more. There was no magic and the relationships were terrible. I'd come to a bad spot in life. It could have been paranoia, but I just didn't feel good – I felt like an outsider.  Ringo, Anthology

Things came to a head during a recording session for Back in the USSR on the 22nd of August, 1968. The precise trigger point is unknown but at some point Ringo snapped. After telling John and Paul he was leaving the group, he walked out of the studio.

At first, Ringo's departure seemed to confirm the underlying reason for it. The others assumed that their drummer's 'resignation' was not seriously intended. They continued the recording session.

Sardinia

The next day Ringo took his family on holiday to Sardinia, where his friend Peter Sellers had a yacht. 
Back in London,the other Beatles that they had a serious crisis to deal with. It proved the jolt they needed to bring them together.

Within days Ringo received a telegram from his band-mates:  'You're the best rock'n'roll drummer in the world. Come on home, we love you.'

Traveller Retuns

Two weeks later Ringo returned to the studio to find his drum kit garlanded with flowers. The effect salutary on all concerned.  'I felt good about myself again, we'd got through that little crisis.'

There were to be many more crises over the next acrimonious eighteen months. George and John would both follow Ringo's example - announcing their departures only to be dissuaded by the others. In the end it would be Paul, Mr Beatl, who would publically announce that the Beatles were breaking up