My favorite examples are the reactions people have when interpreting the Bible. Of course, different groups interpret it differently to bolster whatever arguments they wish, but its stories (or any stories, for that matter) are best understood if the culture in which they were written are understood. Readers of a modern story are bound by their present day cultural biases, but readers of an ancient story are additionally bound by their present era.
At the most basic level, the culture in which one lives might affect one's knowledge of the vocabulary a story uses. This is especially true when a story is written in a completely foreign language with which a reader is unfamiliar. Then the reader must rely on the talents of translators to help him or her understand even the literal meanings of words, let alone their symbolic connotations. Even when one is reading a story written in one's "own" language, vocabulary can be important, as in so-called "local color" stories. Some of Kate Chopin's local color stories, for example, are often difficult for first-time readers to follow because the English they use is so remote from the English with which most of us are familiar today.
Culture influences many things. As a perfect example, the school that I teach at is very rural. Some concepts are hard for my students to grasp based upon their limited cultural ideas. (They simply have been sheltered from the "real world.") Therefore, texts which speak to ideals that they have no concept of are hard for them to relate to and grasp (sometimes).
As for me, I try to read as much as possible so that my own knowledge grows and I can relate to, or understand, texts which are not of my culture.
As has been noted, we all start out in any new situation, whether it is reading a story or meeting a person or visiting a new place, with a perspective shaped by prior knowledge and experiences of your life. So that background obviously impacts the reading process, at least in the initial stages. One of the things I love about reading about different cultures and places is consciously attempting to move beyond my familiar mindset and try to imagine myself in new settings.
Culture influences many aspects of the way we think without a person being necessarily aware of it. When we read, we are exposed to other cultures. How a person interprets that other culture is largely dependent upon their own. I know I had many students who became outraged at the treatment of women in a particular book we read. This is because our society believes women should have equal rights with men. I tried to explain to the students that other societies are not wrong, they are just different. It was hard for them to understand why a woman would be happy and not fight back against a system that did not see her as equal to a man. There was the same difficulty in teaching a story about a young couple that was involved in a courtship rather than a dating relationship. Culture influences what we think of as right and wrong. It also influences our viewpoint on many different areas of life. Reading about a culture that is different from our own can lead to misunderstandings, but it can also open our eyes to a diverse world.
Each reader brings his/her experiences to the page. From this perspective, the text is interpreted, especially if one does not have knowledge of another or older culture. For instance, there are young women of America who find it difficult to understand how Kate Chopin's main character of "The Story of an Hour" has allowed herself to be so repressed in her marriage, for they are unknowledgeable of the Victorian Age and the femme covert laws of that time.
Certainly, readers who approach the narratives of foreign writers need some background in that writer's culture before they can better comprehend the subtleties of the text. By the same token, there is always the human experience in any narrative that is universally understood.
It would not be wrong to say that the culture that we live in is the most determinative to how we read texts and books. The reason for this is that we are all social beings. The very categories of our thoughts are shaped by our culture. So, when we read a text, we will be using the categories of our society. Also the blindspots of our culture will also be our blindspots. The social theories of Peter Berger are excellent to understand how culture shapes the way we think.
The culture that you live in will do a great deal to affect how you understand a story -- what message you take from the story. For example, most Americans would probably read The Giver and react badly to the society. A society that demands such conformity is bad, right? But what if you grow up somewhere in which people are expected to put the needs of the community before their own needs? In that case, you might react badly to Jonas as what you might perceive as his selfishness.
I suppose you could make the same argument about Romeo and Juliet. Instead of seeing them as star-crossed lovers, you might see them as spoiled and selfish brats who shouldn't have gone against their parents as they did.
In the end, people are bound, to a great extent, by their own cultural valences. The culture in which individuals live influences how they read a work because these cultural expectations and norms reflect in their synthesis of a work. The manner in which a culture expresses their own beliefs and ideas impacts in how a particular work sample is revered or reflected. Reading literature is understood through cultural expressions and realities. I don't think that this should be the only way in which works are read, but they are a part of the reading experience. For example, the culture in which one lives will play a large impact in reading a work such as Elie Wiesel's work Night. Reading this work in a Jewish community will be vastly different than reading it in a community that embraces tenets of Aryan supremacy. Reading Tony Kushner's Angels in America will be different in different cultures that have divergent views towards homosexuality and the role of society in constructing such a narrative. The great writer Joyce Carol Oates makes the argument that it is difficult to "transcend" one's culture, explaining why violence is such a part of her writing experience as it is a reflection of the culture in which she lives. In this, one sees how culture plays a role in both the creation and appreciation of literature and how a work is read and understood.