The puritans established a colony in new England in the early 17th century because they?

The basic reason for the Puritan (really should be the sub-sect Separatist) Colony [Massachusetts Bay Colony] as to those settling in the Boston area was religious freedom. Now 'Freedom' is to the beholder at that time.

You wouldn't have been free to worship any other way than then the Puritan way.

There were differences between the Massachusetts Bay Colony (Plymouth) and those who later settled in the areas north of there.

The Bay Colony Puritans were from a sect that held very strict beliefs whereas the later Puritans were considered more 'Liberal' by the standards of those in Plymouth.

Both groups though were looking to practice their religious beliefs the way they wanted without any interference from the King and his Lord William Laud. Laud wanted the Puritans to conform to the High Church (English Church) dogma which included prayers the Puritans refused to say.

It was also a time of great unemployment in England and very high inflation due to the King's policy of printing money and borrowing at high interest. On the King's behalf he did want colonies established in the New World to off-set Spain's 'colonial lead' [like our missile lead in the Cold War].

So basically there were many reasons for the Puritan to leave England in search of better conditions: many more not mentioned here due to space.