As Peter says above, the origin is in the binding of legal documents. However, because the law is always seen as being a stickler for detailed accuracy and nit-picking, the phrase 'red tape' has come to mean any over-fussy administrative practice, usually involving the filling-in of forms in triplicate and so forth ad nauseam, whether to do with law or not. Hence, to 'cut through the red tape' means to develop systems whereby all that nonsense is done away with and processes are simplified.