Mr. Dursley hummed as he picked out his most boring tie for work, and Mrs. Dursley gossiped away happily (с).
Прочитала фразу и нашла разбор понятия почти в половину главы ахаха.
"The word away when it follows a verb as it does there expresses the idea of unrestrained repeated or ongoing action.
When the actor is a sentient being, the idea can be that the actor is "totally immersed" in the activity, that is, not paying attention to what's going on around him or her. When the actor is non-sentient, the meaning is that it was doing the action without interruption, continuously.
They were chatting away.
The baby was babbling away in its crib.
The stew was simmering away on the stove.
To gossip is to relate tidbits of news to another person. To gossip away means either to relate a succession of tidbits about different subjects, tidbit after tidbit after tidbit, or to tell a fairly long story with many details. Usually the news relates to a local person or someone related to them; the story is transmitted by "hearsay". (с)