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.