Museums can inspire people in many ways. If a city has a museum, people have access to out-of-the-ordinary experiences. Whether in a room with sixteenth-century paintings or some moon rocks, being near fascinating objects from the past transports people out of their everyday lives. It can offer them a new perspective or a sense of possibility in their lives.

Andrew makes a good point, but personally, I see greater benefit in the social aspect of museum-going. I think the biggest advantage of having a museum nearby is that people can come together for events like exhibitions and lectures or to volunteer. Museums foster a sense of community in a city.
They can unite people in a common appreciation of something.