Denied unemployment, how do I appeal? (update)

My claim was denied after 23 weeks of waiting, and the letter says I have a short window to appeal. The letter lists a deadline and a hearing process, but it is hard to follow. If you have appealed before, what helped you prepare?

Thanks for writing this out. I went through something close to it 3 months ago, and it settled down once I kept a simple log of every call and form.