Diagnosis
Dishwasher detergent often can't break down baked-on, sticky grease that gets lodged in the tight mesh of the basket. You need a soak with a strong degreaser.
How to Fix It
1) In your sink or a large tub, mix the hottest water you can get from the tap with a generous amount of dish soap (the kind for cutting grease) or a scoop of oxygen-based cleaner (like OxiClean). 2) Submerge the basket completely and let it soak for several hours, or even overnight. 3) Use a soft bristle brush (an old toothbrush works great) to scrub the mesh, following the grid pattern. The soak should have loosened the grease so it brushes away.
If That Didn't Work
For stubborn, carbonized grease, try a boil-out method. In a large stockpot on your stove, boil water with a half-cup of baking soda or a few tablespoons of laundry detergent. Carefully submerge the basket (use tongs) and let it simmer for 15-20 minutes. Let it cool in the water before scrubbing.
Heads Up
Avoid using steel wool or abrasive metal brushes. They will scratch the non-stick coating off the basket. Once that coating is damaged, food will stick even worse in the future. Stick with nylon brushes or non-scratch scrubbing pads.