Yeah it's tricky. You need the whole article to understand what's going on. Also, the language isn't "funny", but the tone isn't terribly serious either. If the topic had been secrets about nuclear power or missiles, it would be more serious, but the secret is just about some muffins, so how serious can it be? : )
That leaves one of the seven, Chris Botticella, out of a job (Chris worked for a company and knew a secret recipe for their famous muffins. He went to work for another company, but the first company brought him to court saying that he can't do that because he will give away the company's secret.....so that's why he's out of a job.)
— and at the center of a corporate spectacle involving top-secret recipe files, (Chris is really involved in this problem. The company wants to protect its secrets and this problem is kind of a "scandal' or "show" .)
allegations of clandestine computer downloads and an extreme claim of culinary disloyalty: (People are talking about the legal case / spectacle, saying that files were downloaded secretly and that someone (Chris) was disloyal to their own food company. I think the word "culinary" is meant to be a little funny because culinary is usually used for really fine food, expensive restaurants, etc. but the food here is just muffins and sweets....junk food really.)
dumping English muffins for Twinkies and Ho Hos.
(He's leaving the Bimbo company, which makes English muffins, and wants to work for Hostess company, which makes Twinkies and Ho Hos. Twinkies and Ho Hos are two popular cake-like sweets.)