Try removing the screens from the windows, then gently wash the screens with soap and water with a rag/car-washing-mit. Sprinkle water to rinse.
My screens use a coated fiber mesh. If yours are not metal, don't use a power washer as it will blow off the coating.
Worse comes to worse, you'll need to get some screening from the hardware store and replace the current screen. There's a rubber ring on the inside of the frames that holds the screening in place. You can easily pop it out of the grove with a dull blade - an old butter knife, a smooth bladed screwdriver, the end of a hacksaw blade (though these have rough edges that need to be smoothed first), as examples.
