Why did the colonies have a representative assembly?
Because they were trying to mimic the English Government.
COLONIAL ASSEMBLIES had their beginnings in the Virginia House of Burgesses, which Governor George Yeardley convened in 1619. After the Sandys-Southampton group gained control of the Virginia Company, they initiated a new policy that provided for a unicameral assembly composed of the governor, his council, and two Burgess to represent each town, plantation, and hundred. Subsequently, the counties, along with certain privileged towns and cities, comprised the units of representation.
In the latter part of the seventeenth century the elected representatives separated from the parent assembly, creating a bicameral legislature. From the start, the Virginia assembly claimed and exercised the right to initiate legislation, and under Governor John Harvey asserted the right to control taxation.
After Governor Sir William Berkeley's withdrawal from public life in 1652 the House of Burgesses exercised great authority with little outside interference except for the limitations the Navigation Act of 1651 placed on commerce. When Berkeley returned to power in 1662, he failed to call elections and retained the old assembly until Bacon's Rebellion in 1676.
Due to popular resentment of his attempt to control the legislature, the assembly reverted to its representative character after Bacon's Rebellion.