It is true that "The Custom House" is an autobiographical short story in itself, where Hawthorne looks back in time and narrates some of his memories. However, this introduction adds substance to The Scarlet Letter by providing a framework within which the reader will see an "old story" unfold under the perspective of the modern eye. Also, "The Custom House" is what basically leads the reader to understand how the story of Hester Prynne came to the narrator's ear.
It is common knowledge that Nathaniel Hawthorne, in adding "The Custom House" toThe Scarlet Letteras an introduction, did it mostly to extend the length of the narrative and also to lighten up the main idea, since the novel was sure to cause a reaction in his intended audience. As a result, many students wonder what is the worth of reading "an added piece" that neither adds nor takes away from the sequence of events that occur in the novel; it is an appendix of a sort which completes and enhances the narrative.
However, in order to experience the dramatic and literary contrast between the old and the new (new, meaning 19th century standards versus those of the 17th century where the novel is set) "The Custom House" is quite recommended to be read.
As it starts, "The Custom House" is told in first person (presumably it is Hawthorne himself speaking) and tells the story of how the narrator, who once successfully published his experiences when he used to live in an Old Manse, was asked to write again about his experiences working at the Custom House.
As added details, Hawthorne describes the salient traits of some of the people whom he remembers the most, up until his labors at the Custom House cease, giving him the time to put together the curious story that he had been told about Hester Prynne. This device is quite a clever way to add realism to the story, and to move it from fiction to (almost) folklore.
Hawthorne also builds momentum forThe Scarlet Letterthrough "The Custom House" because he explains how he came about the scarlet letter itself, which he found stuffed between a number of other items. The narrator imagines that these items that he found used to belong to a time where "something huge" must have occured that compelled the safekeeping of such a token as a scarlet letter, which by now has lost what seems to once have been glitter and ribbons.
Finally, Hawthorne warns the reader about the dangers of labeling and judging others, reminding us how likely history is to repeat itself if we do not correct the mistakes that slowed our path to greater knowledge.
Therefore, it is up to the reader whether they wish to skip "The Custom House", however, an advice would be to at least read it, even if it is after reading the novel, because you may experience the "full circle effect" once you read about how this unnamed narrator "stumbled upon" Hester's story.
According to Hawthorne scholars, Hawthorne wrote the actual story of The Scarlet Letter first. He added "The Custom House" sketch after completing the narrative itself. One reason for adding The Custom House is that Hawthorne felt that the story of TSL was too dark and somber for a wider audience. His reason for adding The Custom House sketch was to balance his dark tale of guilt and punishment with a lighter touch.
A second reason for adding The Custom House sketch is that Hawthorne and his publisher worried that the original narrative of TSL was too short to market as a stand-alone novel. Hawthorne's addition of The Custom House sketch was an attempt to pad out the length.
So, in answer to your question, you probably don't need to read The Custom House sketch to understand TSL. However, once you've read The Custom House sketch and TSL together as a single work of art a few times over, you may--like me--find yourself agreeing with Hawthorne's artistic judgment. The Custom-House and the rest of The Scarlet Letter complement and balance each other on the level of tone. You get a much better sense of Hawthorne's range as a creative artist. Reading TSL without The Custom House just wouldn't be the same emotional experience.