Answer: Forcing colonists to allow British soldiers to stay in their homes.
Quartering of soldiers meant soldiers would have the right to enter your home and expect you to provide them food and lodging. That was seen by the colonists as invasive of their personal rights.
The issue of quartering of soldiers appeared not only as a complaint in the Declaration of Independence (as quoted in the question), but also became a featured right in the Bill of Rights of the Constitution. The 3rd Amendment states: "No Soldier shall, in time of peace be quartered in any house, without the consent of the Owner, nor in time of war, but in a manner to be prescribed by law."
Court justices have used the 3rd Amendment statement about the quartering of soldiers to show that the constitution does give attention to citizen's right to privacy, even if not using the exact term "right to privacy."
