Strictly speaking, it wasn't. The United Kingdom got along quite happily without one. The problem was that the Founding Fathers had done away with the monarchy which was the centre of British democracy. All power in the UK was formally vested in the Crown. It was to the Crown that judges, politicians and the military owed their allegiance. No party or faction could take over the reins of power completely without overthrowing the King or Queen, which was exactly what had happened in England under Cromwell and had just happened in America.
The Constitution therefore became a contract between the governed and the governors, setting out the rights and duties of each to the other and establishing the separation of powers between the Executive, the Congress and the Judiciary.