The argument for a 'level playing field' sounds like an argument for simple justice.
International trade is a game played on a sports field.
If the field slopes in one direction, one team has an unfair advantage. The field is thought to be tilted, for example, by a state that provides a subsidy to an industry, allowing it to reduce the price of its exports. Since states can't change ends every quarter the solution is to do away with these subsidies. In practice 'Subsidies' are very broadly defined to include things like regional development programs, generous unemployment insurance benefits and even publicly-funded health care. Trading partners can argue that because these benefits are provided by the state, employers receive a hidden subsidy.