The story
Abu Yusuf Yaqub ibn Ishaq al-Kindi worked in Baghdad in the 9th century, when the city's House of Wisdom was translating Greek philosophy, mathematics and medicine into Arabic. He wrote on music, optics, medicine and astronomy, and some hundreds of other treatises. One of them, A Manuscript on Deciphering Cryptographic Messages, survives in a copy in an Istanbul library and came to wide attention only in 1987, when it was published.
In it, al-Kindi explains that to solve a cipher in a known language you should find a long sample of ordinary text in that language and count how often each letter occurs. Then you count the signs in the cipher. The commonest sign is probably the commonest letter, the second commonest the second, and so on, until enough of the message shows through to guess the rest.
The method seems to have come out of religious scholarship: to date texts and check whether sayings were genuine, scholars studied the words and letters of the Qur'an and the hadith closely, and noticed that some letters are simply more common than others.
Why it worked for so long
A simple substitution has 26 × 25 × 24 × … × 1 possible keys, about 4 × 1026. No one could try them all, then or now by hand. But a frequency count does not try keys at all: it reads the statistics of the language through the disguise. Short messages are harder, because the counts are noisier, which is why this board's message is a long one.
How it was broken
Frequency analysis broke every cipher that used a single fixed alphabet: Caesar shifts, Atbash, keyword alphabets, and the symbol sheets of Renaissance courts. Europe's cipher secretaries knew it well by the 1500s. Defenders answered with tricks: dummy signs that mean nothing, several signs for E, code words for common names. The real answer, as the Vigenere platform shows, was to keep changing the alphabet.
The trick has limits. A deliberately odd text breaks it: in 1969 the French writer Georges Perec published a whole novel, La Disparition, without a single letter E.