Diagnosis
Food grease and debris have baked onto the basket rails and the sliding channels inside the fryer. This acts like glue.
How to Fix It
Unplug and let the unit cool to room temperature. Pour a few tablespoons of white vinegar into the bottom of the fryer bucket, around where the basket sits. Let it soak for 30 minutes. The acidity breaks down the grease. Put on an oven mitt, grip the basket handle firmly, and twist it side-to-side while pulling out. Do not yank straight out.
If That Didn't Work
The basket release button mechanism may be jammed with debris. Look at the button on the basket handle. Press it repeatedly while wiggling the basket. If it's stuck in, use a thin tool like a toothpick to clear gunk around it.
Heads Up
Clean the basket rails and the interior channels with a damp cloth after every 2-3 uses to prevent this. Do not use cooking spray directly on the basket.