Why do most question words in English begin with wh? The answer involves a hypothetical ancient language, a semantic root shared with cheese, and curious people throughout the ages.
To explore why we say why (and who, whom, whose, what, when, where, whether, which, whither, whence, whatever, whoever, and wherefore!), we first need some backstory on the English language.
English is part of a large group of languages called Indo-European. Within Indo-European are many subcategories, including Germanic languages (of which English is a part) and Romance languages (from which English has borrowed heavily through French and Latin). English is most closely related to other members of the Germanic language family, including German, Dutch, and Frisian.

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Now, enter historical-comparative linguistics. In this field of study, linguists attempt to reconstruct the ancestors of currently spoken languages by comparing similar words in related languages and proposing words that may have preceded them. Proto-Indo-European is a reconstructed language that represents the ancestor of the Indo-European language family.
Interestingly, most of the question words, or interrogatives, in English are derived from the same Proto-Indo-European root *kwo- or *kwi-. (The asterisk next to the word indicates that the word is a hypothetical reconstruction and has not been documented in natural language.) This root word, which likely meant “who” or “what,” then formed the stem for all interrogative pronouns. (Interestingly, this word is also the root of words as diverse as cheese, neither, and quote.) The sense of asking about a person or a thing was extended to asking about a place, a time, or a means of doing something. Different forms emerged to fill the speakers’ need to ask different types of questions. Though modified in form, these question words are now found in languages that descended from this prototypical language. Within many Indo-European languages, these words still display an etymological relationship with each other through a shared initial sound.
Linguists assume that regular sound changes occur over time, producing systematic differences in languages descended from a common ancestor. The most well-known set of sound changes in Germanic languages is known as Grimm’s law. As part of this set of sound changes, the k- sound in *kwo- and related words shifted to a x- sound (this sounds kind of like clearing your throat).
Further spelling and sound changes caused a shift from x- to hw-, and hw- was reduced to simply w- or h-, depending on the vowel that followed. As part of a systematic spelling change in the Middle English period, the spelling of most question words reversed the hw to wh.
And how about how? The word how in Old English was hū, which may have originally been *hwu. You can imagine how difficult the w- and u- sounds are to say without condensing them into one sound. The condensing of the w- and u-is fairly common because the w- sound is very similar to an elongated u- sound (oo). The spelling of how followed suit.

Let’s take a quick look at other languages. In German (a close relative of English), question words are wer, was, wann, wo, warum, and so on—even wie, the word for how, begins with a w. German mirrors English in the development of interrogatives beginning with wh, though the h was dropped from the spelling of those words in German.
In Italian (less closely related to English, but within the Indo-European category), sound change stayed closer to the original *kwo-. Most interrogative pronouns begin with a k- sound: chi, che, cosa, quando. Notably, some have departed from this root: dove (where) and perché (why).
Likewise, in Sanskrit (another member of the Indo-European language family), all the interrogative words have retained their initial k- sound: kaha (कः), ke (के), kaa (का), kim (किम्), kadaa (कदा), kimartham (किमर्थम्), and and kutra (कुत्र), among others, all show evidence of descent from the Indo-European *kwo-.
The story is the same throughout the language family: one common ancestor word gave rise to a variety of question words that follow a similar pattern in different languages. And now you know why we say why!
Sources
“*kwo-.” Etymology Online. Accessed November 10, 2020. https://www.etymonline.com/word/*kwo-.
Lyons, J. “Linguistics.” Encyclopeadia Brittanica. https://www.britannica.com/science/linguistics#ref35069.
“Reconstruction:Proto-Indo-European/kʷos.” Wiktionary. Accessed November 10, 2020. https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/k%CA%B7os.
San Filippo, Michael. “How to Ask Questions in Italian.” March 11, 2019. ThoughtCo. https://www.thoughtco.com/how-to-ask-questions-in-italian-2011117.
“Simple Interrogative Words: Who, What, When, Where, Why, and How.” Learn Sanskrit Online. Accessed November 10, 2020. https://learnsanskritonline.com/lessons/Simple%20Pronoun%3A%20He%2C%20She%2C%20It%20and%20They%20/simple-interrogative-words-who-when-why-where-and-how#:~:text=We%20have%20learned%20few%20interrogative,’who’%20in%20feminine%20form..
“what.” Etymology Online. Accessed November 10, 2020. https://www.etymonline.com/word/what.