When did the Beatles start and breakup?

We'll start with the time period - it probably started in 1968 during the White Album; Ringo walked out of the band. He was persuaded to come back, early in 1969 George left the band - he was persuaded back to finish Let It Be and to record Abbey Road. After Abbey Road, all four knew the game was up.

The "why" is more complicated; however, Paul's explanation is probably the best - it was like being in the army, at some point you have to stop being with your mates all the time, you grow up, get married and start living normal lives - that's what happened to The Beatles.