DIAGNOSIS That cloudiness is usually a combination of fine surface scratches from everyday washing and mineral scale (hard water deposits) etched into those scratches. It's cosmetic, not a functional problem.
FIX For mineral scale, a good soak with a mild acid is your friend. Make a solution of one part white vinegar to three parts warm water. Fill the pitcher and let it sit for an hour. Empty it, then wash thoroughly with mild dish soap to remove any vinegar smell. For the fine scratches, you can try a very gentle plastic polish or a paste made from baking soda and water. Apply it with a soft, damp cloth in small circular motions, then rinse.
IF THAT DIDN'T WORK The cloudiness might be permanent if the plastic has been etched by harsh chemicals or cleaned in a dishwasher. Dishwasher detergent is particularly abrasive and hot water can warp and dull the plastic. Unfortunately, there's no fix for that kind of deep damage.
HEADS UP To prevent this going forward, always hand-wash the pitcher with a soft sponge or cloth. Avoid abrasive scrubbers like steel wool or harsh scouring pads. Never use bleach or strong cleaners like ammonia, as they can chemically damage the plastic and make it brittle.