What, in broad terms, did the British do, both governmentally and socially, that discouraged some of the American colonists from their loyalty to the British empire?
Broadly speaking, the British attempted to restrict what the colonists came to understand as their natural rights. The Colonial Era coincided with the Enlightenment, an era of expanded education and reoriented philosophical beliefs. So, at the same time that King George was in England limiting trade and enforcing taxation, Jean-Jacques Rousseau was in France publishing that all men are born equal and free. And that is when Thomas Jefferson entered the stage of world history and wrote the Declaration of Independence.
In fact, if you look at the Declaration of Independence, it includes a clearly delineated list of grievances against the king. Here are a few examples:
1) Taxation without representation: the British imposed taxes on the colonists without giving them a say in government. They taxed things like tea and stamps.
2) Restricting territorial advance. The colonists very much wanted to expand westward, but the British forbade expanding west of the Appalachian Mountains in the Proclamation of 1763.
3) The British required colonists to house British soldiers.
4) The British limited trade between the colonies and other countries.