Badeye
It's one of the many things that is influenced by the Japanese awareness of personal and public space. Many Japanese might find it uncomfortable to see or hear someone eating in a public space, and therefore feel uncomfortable if seen eating. In a restaurant or other space designated for eating, everyone is doing the same and entered the restaurant knowing there'd be eating going on, so it's no biggie. But out on the street or in a public space which everyone has to pass through, accidentally getting stuck with someone eating could feel uncomfortable. People's eating habits (such as if they are noisy) and things like the messiness of food could be a disturbance. I think it also has to do with peer pressure- if everyone is doing it, it's ok. If it's just one person, you'll stick out like a sore thumb.
An exception to this seems to be on public transport. It's not so often, but on local buses sometimes I see people eating (and will have a snack, myself) and on trains it's entirely common for people to have drinks and snacks, even full bento lunches. This is likely because a person stuck traveling all day doesn't have the option to wait to eat elsewhere.
There are people who will eat instant ramen in front of convenience stores and so on, but they're often looked at as being uncouth and inconsiderate of other people's space.
Bottled water or tea are fine and onigiri (riceballs) are alright if eaten with discretion. If you are in a tight space with someone else, such as sitting next to someone on the bus, it's a bit old fashioned but you can excuse yourself with "Shitsurei shimasu" (I'm going to be rude, excuse me) before eating.
I'd like to mention a few other things about eating.
Chopsticks (hashi)
Eating at a restaurant or in someone's home, you'll likely be given disposable chopsticks (waribashi). When breaking apart waribashi, it is most polite to hold them horizontally and pull up and down. This is to keep your arms from swinging out to your sides and potentially bumping your neighbors.
NEVER stick chopsticks straight up in food. Japanese funerals customarily place chopsticks straight up in a bowl of rice. Similarily, never pass food from chopsticks to chopsticks, as it resembles the passing of cremated bones at a funeral.
When not using your chopsticks, you can place them together across your dish or rest them on a hashioki, or chopstick holder- a small, generally plastic or ceramic item used to keep the eating end of the chopsticks up off the table.
If you are in an extremely formal dinner situation, you should serve yourself from any group plate with the back end of your chopsticks (the side you're not eating off of). You should also avoid gesticulating or pointing with your chopsticks.
Showing respect
Before the meal, it is customary to say, "Itadakimasu". It means to humbly receive, and is a show of appreciation to your host, the cook, the animals and plants, even the sun and rain that grew the rice. It is a gesture of appreciation for the meal.
If after the first round of your meal, new food is served to you, when accepting it you should again say "itadakimasu". This also goes for when desert is brought out. If in doubt about when to say it, watch the people around you and follow their example.
When you have completely finished eating, "Gouchisosama deshita" (It was a feast) shows your appreciation. If eating at a restaurant, you can also tell the tenant "Gouchisosama deshita" as you pay your bill or leave. It's a bit old fashioned but will not go unnoticed or unappreciated.
While eating
This is a rather large and pretty fancy 'teishoku' meal (traditional Japanese style meal)
Appreciate that a traditional Japanese meal is made with balance in mind and eat a little of each item at a time rather than just powering through the rice, then eating all the main dish at once, and so on. A bit of rice, a few sips of miso soup, a bit of fish, a pickle or two, and so on. Most meals are prepared in a way that the separate dishes compliment each other and the flavors are always changing. This keeps things new and helps if there is something you really don't like as you can combine it with other flavors to help you get through it. It's great if you can finish all of the different dishes about the same pace.
The best way to eat is to lift each small dish as you pick things up off of it with your chopsticks. For soup, you drink directly from the cup.
Cheers (Kanpai)
In formal or party situations, there will likely be a toast. Don't drink before everyone says Kanpai. After you say Kanpai, you take a two or three second drink and then everyone will clap. There will likely be one kanpai at the beginning of the party to get things started and one at the end to finish things out, so leave a little drink left in your glass.
It's customary for a subordinate to fill their superior's glass as soon as it seems to be getting low, and they will not stop, ever. If you don't want anything more to drink, simply don't drink from your full cup. It's a clear sign you don't need a refill.
Feel free to ask any other questions you have about food, eating etiquitte, or table manners here.
