The English colonies in America developed ideas about self-government from three main sources. First, there was their history as English people. Second, there were the ideas of the Enlightenment. Finally, there was their experience during the period of "salutary neglect."
First, English people had been developing ideas about self-government ever since at least the Magna Carta. This had evolved over the centuries and can be said to have culminated (at that time) in the Glorious Revolution of 1688. The colonies were heir to these ideas and had a clear sense of their rights as Englishmen.
Second, they were influenced by the ideas of the Enlightenment. The elites of the colonies had read thinkers such as John Locke. From those sorts of thinkers, they absorbed the idea that government drew its legitimacy from the consent of the governed.
Finally and more practically, the colonists had had a great deal of experience governing themselves. For decades, the British government left them more or less to themselves, allowing them to develop the idea that they should be autonomous.
From these sources, the colonists derived the idea that they should be self-governing.