What does this quote mean?"What lasts is what is written, we look at literature to find the essence of an age."
There are two parts to this quote, each with a bit of a different meaning.
The first part of the quote is saying that only things that are written last. It is saying that things that are spoken do not last because they are gone once they are said.
The second part says that some of this written stuff (literature) can tell us what things are like during a particular time and at a particular place. It is saying that literature can tell us what kinds of things were worth discussing in a given time and place.
I notice that you tagged this with "The Scarlet Letter." In this case, you could say that Hawthorne's novel shows what people of his time were concerned about. In his case, it would be with issues of morality and conformity.
Literature or the written word becomes the document of history. Some books are timeless such as "Little Women" and Jane Austen's novels. When we read a book we see the past in it. The style of writing, the illustrations expressed through words, the small details, all give way to the period which the novel represents. Some literature is written to document history verbatim. However, whatever is written has taken on its own life and journey.
Humans only live for a period in life, but literature continues on long after the writer has laid his pen down and decayed in the grave.