Why did the Pilgrims leave Europe?
For religious freedom.
In England the state religion was Anglican. The Puritans and Pilgrims were not part of the state religion. The political power of the government was used to enforce religious conformity. As non conformist these religious groups suffered persecution.
The Pilgrims had immigrated to Holland to escape persecution but feared the loss of their English heritage living in a foreign country.
Purchasing a charter to establish a colony in the New World would enable the Pilgrims and Puritans to set up their own society where they would be free to worship as they pleased. By leaving Europe and the religious oppression experienced there the Pilgrims hope to find religious freedom.
Most of the passengers on the Mayflower wanted a better social and financial position than they could get in England.
The religious separatists, or Puritans, are the Mayflower passengers that have grabbed history's attention. Of the 102 passengers aboard the Mayflower, 40 could be described as "Puritans" (The term "pilgrim" came much later), although quite a few of the remaining 62 were servants or other employees of that core 40.
The Puritans tried to set up a colony in Holland, but feared losing their English culture, so a group of 40 of them resolved to go to Virginia, where several cities had begun along the James River. They would have been a religious minority there, too--Virginia colonists were primarily there to make money--but that was the plan, nonetheless.
Bad weather and worse navigation brought them instead to Plymouth Bay (modern-day Massachusetts). They might have pressed on to warmer Virginia--they got there at the beginning of winter--but they ran out of supplies, including beer, a low-alcohol dietary staple on ships in those days when potable water was hard to come by.