I do eat at my desk sometimes, and I don't mind when my employees do it. But we're a small office, only 3 of us, and no customers coming in. I can see where there are some concerns with it.
One is computer equipment/messes. Most adults can manage to eat without getting 5 tons of crumbs in the keyboard, but I can see where a large company might have to start this policy because of a couple of pigs getting spaghetti in the printer, or getting stains on important documents, etc.
Another is, if people are going to continue to answer the phone while eating. Again, most of us here, I would think, know better than to answer the phone with a mouth full or smack into the phone but- the lowest common denominator can be the reason for a lot of these rules.
Ideally all workplaces should provide a place to eat. Even though we don't have room for a separate dining area I made sure to buy a microwave and refrigerator so people can bring their lunch. It helps our pocketbooks and helps us keep healthy eating habits, not having to eat out every day. My office is huge and I've told my people they can eat at the conference table in my office if they want to just get away from their desks to eat. No one has taken me up on it, but they do have the option.
I always hated when I'd bring lunch back and eat at my desk and no one recognized I was on break and would still expect me to answer phones. One of my employees will go grab a sandwich and bring it back to eat and still have time left on his lunch hour. I make sure to grab the phones so he gets his actual break. I don't even know if it's a peeve of his, but I'm very conscious of it because of past experience.