Got a denial letter for unemployment, what now? (update)

My claim was denied after 4 weeks of waiting, and the letter says I have a short window to appeal. Thank you in advance. My employer's version of events does not match mine, and I have emails that show it. If you have appealed before, what helped you prepare?

Thanks for writing this out. I went through something close to it 2 months ago, and it got easier once I kept a simple log of every call and form.

I printed my emails in order and wrote a one-page timeline. The hearing officer kept going back to it.