There are many reasons to do public engagement but many fall into three categories:
The Moral Case
Universities receive public funding and most researchers are funded, at least partly, by the public purse. This means that there is a duty to ensure that the knowledge and understanding gained by the researchers is spread as widely as possible so that as many people can use the new information, products or techniques.
The Business Case
It is clear that the funders of universities and the Government want universities to demonstrate their societal impact. For many academics this is a contentious issue, but at its simplest it is making sure that the groups of people who you know may be interested in a new finding or product are able to learn about that and that the information is not only published in academic journals where many people cannot find them. This could involve adapting academic articles for trade publications.
The Academic Case
Universities actively contribute positively to society in many ways. Engaging academics and society encourages curiosity and helps promote a desire for formal and informal learning. For researchers, working with wider society can help to identify new problems to solve or new uses for existing knowledge, or products.