Why is it important that artists feel as if they can express themselves? What does art say about the culture and society in which we live?

The notion of art as a form of self-expression is relatively recent. For many societies, art is a way of encapsulating and handing down cultural norms, and serves to foster a sense of community rather than individualism. Thus rather than saying that it is important for artists to "express themselves" it might be better to think of this formulation in terms of the second sentence of the question, namely how theories of art that emphasize individual self-expression reflect specific societal norms.
In oral traditional cultures, for example, literary works tend to be anonymous and collaborative with individual bards making gradual and minor adjustments to works they inherit from their predecessors and hand down to their successors. Many forms of architecture and art also are created collectively and meant to express a communal ethos rather than to glorify the individual. Such forms of cultural production suggest a society that values collaboration and tradition over individual ego.
The romantic notion of art as individual expression suggests a society in which community has fractured and the isolated individual is glorified. Rather than works of art being seen as contributing to the common good and expressing shared values, they are seen as a form of self-gratification or a rebellion against existing social norms.
While it is important that artists not be censored, because freedom of expression allows for the important function of social critique and challenging accepted beliefs and ways of thinking, that is different from art simply being a matter of "self-expression" or egotistical self-indulgence.