Short answers, pulled from the story.
The Beatles travelled to Rishikesh in February 1968 to study Transcendental Meditation at the ashram of Maharishi Mahesh Yogi. Their interest followed their denunciation of drugs in favour of TM, and was led by George Harrison's commitment to the Maharishi's teachings.
Estimates range from 30 to 48 songs written during the Beatles' 1968 retreat in Rishikesh. Eighteen of those songs appeared on The Beatles (the White Album), and two others, "Mean Mr. Mustard" and "Polythene Pam," appeared on Abbey Road.
The four Beatles left at different times and for different reasons. Ringo Starr left on the 1st of March after ten days, citing the food and insects. Paul McCartney departed in mid to late March for business reasons. John Lennon and George Harrison left abruptly on the 12th of April following rumours of the Maharishi's inappropriate behaviour toward Mia Farrow and another female student, suspicions encouraged by their associate Alexis Mardas.
While waiting for taxis to leave the ashram on the 12th of April 1968, Lennon wrote a song he called "Maharishi," with the lines "Maharishi - what have you done? / You made a fool of everyone." The song was later retitled "Sexy Sadie" before appearing on The Beatles.
The Maharishi's International Academy of Meditation in Rishikesh was eventually abandoned. As a result of continued interest in the Beatles' 1968 retreat, the compound was opened to the public in 2015 and has since been renamed Beatles Ashram.
George Harrison and Paul McCartney both reconciled with the Maharishi. Harrison gave a benefit concert for the Maharishi-associated Natural Law Party in 1992 and publicly apologised for the way the Maharishi had been treated. McCartney visited the Maharishi in the Netherlands in 2007, renewing their friendship. By the time of the Maharishi's death in 2008, both had stated that allegations of sexual impropriety against him were untrue.