The meaning of the American phrase "and all that jazz" is basically the same as the Latin phrase et cetera, commonly abbreviated "etc." and used at the end of the listing of a series to indicate that there are more related items that are not specifically listed. The phrase is often used in a dismissive manner to indicate that what follows is unimportant or silly in nature. As the origins of idioms go, it is a very modern phrase.
Rather than being an idiom or proverb, "and all that jazz" is probably better explained as a slang phrase. The word "jazz" is a common slang term. Common phrases using the word include "jazzed" and "jazz up," among many other variations.
The word "jazz" probably entered the English language from a Creole phrase that referred to sex and a type of dance. There are several theories about the etymology of the word, but the definite origins remain undetermined. "Jazz" was first used to refer to a type of music, ragtime originally, in about 1913. The phrase "and all that jazz" didn’t come into usage until about 1939.
Although probably already in use to some extent, the use of this phrase meaning to continue a series in a somewhat dismissive manner probably entered common language as a result of the 1975 musical Chicago. Based on a play written by Maurine Dallas Watkins in 1926, the musical's opening number is titled "All that Jazz" and used the phrase in the manner of meaning et cetera. The song is performed by Chita Rivera in the original cast album from the musical, but Liza Minnelli's 1975 single of the song gave the phrase even more popularity. The song begins with the lyrics "Come on, babe / Why don’t we paint the town? / And all that jazz." The phrase is repeated throughout the song's lyrics.
The use of the phrase originated in the United States, though it is now commonly used in everyday language in Britain and Canada. The phrase also lends itself to the title of a 1979 musical, an Ella Fitzgerald album, and a radio series, among other uses in literature and music. "And all that jazz" is often used, sometimes as a pun, in marketing arts and music stores, events, and festivals.