Good manners are more than just nice. Like social traffic laws they take some of the guesswork out of interacting with one another. But perhaps more important is that good manners acknowledge the fact that no matter how brief an encounter is, there's a fellow human being on the other side.
Manners and etiquette have a practical value. Knowing in advance who is expected to go first or have a seat keeps us from having to negotiate or fight it out every time. Saying "thank you" marks the successful passing of a favour from one person to another and brings the exchange to a close. "Please" and "excuse me" request permission to come into someone else's territory. A world without good manners would certainly be confusing and potentially brutal. Many of our customs are there to protect groups who would lose out if advantage were left to the swift or the strong. Good manners provide guidelines to follow in times of uncertainty and reduce the need of awkward exchanges and the risk of confrontation.
But good manners do more than make the social world easier to navigate. Our "pleases" and "thank yous" show that that we're aware of dealing with another human being rather than an object. This is why poor manners can feel so offensive. It's not like a direct insult which is personal. Lack of manners offends because it's impersonal. It fails to recognize that a two-way connection is taking place and sends the message "I am not aware of you as a person".
One of the reasons traditional courtesies may have slipped in recent years might be that we spend so much time interacting with non-human things. You don't have to ask a television to switch channels or thank a computer for recording your files. People who neglect manners, however, do so at their social and professional peril. Manners demonstrate an awareness of one's social environment. Even if no serious offence is taken, poor manners can still make someone appear less engaged in what's happening around them and more caught up in their own world. In other words, good manners connect people to those around them, poor or bad manners can cause a separation.
Imagine, for example, there is a small group of loud-talking cinema-goers disrupting the film for the rest of the audience. The auditorium is now split into an "us-and-them". Whether they're aware of it or not, the rowdy group has become isolated from the rest of the cinema community.
Good manners are a code of common practice. What constitutes good manners can differ from group to group or culture to culture. Whatever the customs are, respecting them shows a desire to actively engage with a community and the people in it. Failing to adopt them sets you apart. We might forgive people who let their manners lapse, but we'll feel more positive about and strongly connected to those who don't.