why was mayflower compact established?

Answer 1

One piece of this that isn't always mentioned is that 'this iw the way the Puritans did it' --- the compact was an extension of the sort of 'church covenant' arrangements they used for the governing of the local church in each community. (This was the "congregational" system later dubbed "the New England Way", and an interesting early example in practice of the whole nation of "government by the consent of the governed", which fed into the ideals of the American Revolution and founding of the new republic.)

As for the specific practical reasons:

Part of the explanation is found in the document itself. Here's the meat of it -- note esp. the parts I've placed in CAPITALS:

"for the Glory of God, and Advancement of the Christian Faith, and the Honour of our King and Country, a voyage to plant the first colony in the northern parts of Virginia; do . . . , covenant and combine ourselves together into a civil Body Politick,
FOR OUR BETTER ORDERING AND PRESERVATION, AND FURTHERANCE OF THE ENDS AFORESAID;
And by Virtue hereof TO ENACT, constitute, and frame, such just and equal LAWS, Ordinances, Acts,Constitutions and Offices, from time to time, as shall be thought most meet and convenient FOR THE GENERAL GOOD OF THE COLONY . . . "
http://www.njstatelib.org/Research_Guides/Historical_Documents/HISTDOC1.html

BUT there was another reason why William Bradford, et.al. decided the document was NECESSARY. There were many others besides the Puritans on the ship, and there was concern about their going their own way. So they needed to be 'reined in' by committing to the community.

Bradford's own history of the colony explains this. Here is a summary (from the page linked below):

"The Pilgrims, reinforced by some seventy “strangers” from London, sailed for Plymouth in September 1620 and arrived off Cape Cod in November. They missed the coast of Virginia. Some of the London recruits were a discontented, “undesirable lot” and made “mutinous speeches.” Bradford writes that the “strangers” boasted that they were not under the jurisdiction of the Virginia Company and “would use their own libertie, for none had the power to command them, the patent they had being for Virginia and not for New England…”

(William Bradford, Of Plymouth Plantation-1620-1647: A New Edition: The Complete Text, with Notes and an Introduction, Samuel Eliot Morison, Alfred A. Knopf, New York, 1993, p. 75)

"Since the patent, or charter, was only good in Virginia,
some form of government had to be established if the settlers
were to maintain peace. The Pilgrim leaders drew up the
Mayflower Compact. . . ."
http://www.americanheritage.org/AHEF_Texas_High_Extraction_07_3_MayflowerCompact.pdf

Answer 2

In short, because on the trip to America, the "pilgrams" realized that they would need some form of rules and laws to keep their settlement civilized.