r/AskHistorians probably isn't the best sub for this question, but the answer is conjugation. Before answering your question in more detail, I would point out that what you heard on the radio was not representative of language during the Middle Ages (moreover the Middle Ages covered a very long period and language was not consistent throughout). What you heard would be more consistent with the 15th or 16th century. One of the most common places people encounter this kind of language would the King James Bible. The explanation below is for that era of language, and although the gist of it applies to the Middle Ages--i.e. "conjugation,"--the details would be different.

"-eth," and "-est," (and its variations) are suffixes to conjugate verbs. "-eth" and "-th" for irregular verbs are used for third person: "She shareth her wealth." "He hath one goat." "-est" (also sometimes "-st" or "-t" for irregular verbs) is used for second person: "Thou sharest thy wealth." "Thou art a true friend." Conjugation gets more complicated when you introduce tense and plural verbs. If what you heard on the radio put the suffix on anything other than a verb, it was just wrong, and would explain why it sounded like nails on a chalkboard. This has become popular lately as a form humor, kinda like "DoggoLingo."

This comes from how verbs were conjugated in Middle English. -eth was a common marker of the 3rd person present tense singular. I'm sure you could get a more in depth answer about how this in particular became culturally associated with "Middle Ages speak" in general, however.

Source: Thomas Pyle's and John Alego's The Origin and Development of the English Language