The Mabinogion is made up of 11 tales set in a magical landscape featuring magical white horses and heroic men.
Matthews visits the Bodleian Library in Oxford to examine the 14th Century manuscript - The Red Book Of Hergest - which includes four branches of the stories.
The documentary on BBC Four is part of The Secret Life of books series.
The tales include one of the earliest-known references to King Arthur.
Some of the stories are more than 1,000 years old but others are thought to be older still.
Their influence can still be seen in inspiring modern day fantasy fiction, and big and small screen hits like Lord Of The Rings and Game Of Thrones.
Matthews traces the history of the tales, which owes much to Victorian aristocrat Lady Charlotte Guest, whose translations helped bring the Mabinogion to a modern audience in the mid 19th Century.
Lady Charlotte, who had taught herself Arabic, Hebrew and Persian, married south Wales industrialist John Josiah Guest and learnt Welsh.
The "Red Book" is the most important Welsh book to survive from the medieval period.
Written in Middle Welsh it may have been an act of cultural preservation at an uncertain time.
Prof Thomas Charles-Edwards of Oxford University said: "This was a strong period for grand, local Welsh families patronising literature and possessing manuscripts as things of pride.
"This was written for a particular patron who lived near Swansea Hopkin ap Tomos and he had a wide knowledge of Welsh tradition, this manuscript corresponds to quite a lot of his range of interests."
Matthews also reads extracts of the stories, which are set in a magical landscape, connected to west and north Wales.
This is a land where white horses appear magically, where a giant King can stride across the sea, and there are beautiful women and heroic men.
"When I was growing up I had a Mabinogion poster, the illustrations were almost psychedelic," said the former lead singer with Catatonia.
"I'd fallen in love with the unpredictable plots, the beautiful language and the larger than life characters".
The Secret Life of Books  remains available on the BBC iPlayer until 7 October.
Singer and radio presenter Cerys Matthews is sharing her personal passion for a collection of Welsh folk stories dating back centuries.