Diagnosis
Years of built-up grease and grime have essentially glued the aluminum mesh filter into its slot. It's a very common issue in over-the-range units.
How to Fix It
Don't force it and bend the frame. Apply a degreaser. Spray a kitchen degreaser or a mix of hot water and dish soap along the edges where the filter slides. Let it soak for 10-15 minutes to soften the grease. Then, wearing gloves for grip, try wiggling it side-to-side while pulling firmly downward. Sometimes tapping the frame lightly with a rubber mallet can help break the seal.
If That Didn't Work
For a truly seized filter, you may need to carefully pry it. Use a thin putty knife or a stiff plastic card. Gently work it into the gap between the filter frame and the microwave body, and slowly twist to create leverage. Work your way along both sides. It will pop free.
Heads Up
Once you get it out, clean it thoroughly and, before reinstalling, wipe a very thin film of cooking oil or spray a dry silicone lubricant on the filter's sliding edges. This will prevent it from sticking again next time. Never use a sticky grease like Vaseline.