Why は is pronounced as 'wa' when it is used as connecting word

If you're curious about the actual historical reasons:

The consensus is that は{行|ぎょう} was originally pronounced with a P sound in Old Japanese. So, {花|はな} was originally pronounced pana. The P sound involves pressing one's lips tightly together to abrupt the airflow, without the vocal folds vibrating.

But with time Japanese people increasingly started pronouncing the P sound with the lips slightly parted, such that the airflow was not blocked completely. This produces a sound kind of like blowing out a candle, it's a bit F-like which is why it's usually represented with that letter in romanizations. This sound change was often blocked by ん and little っ, which is why aside from onomatopoeia and foreign loan words, one only really finds ぱ{行|ぎょう} after ん and little っ in words like {散歩|さんぽ} or {切腹|せっぷく}. In fact this is also why ぱ{行|ぎょう} has that unique ring diacritic: the {半濁点|はんだくてん} was invented by Portuguese missionaries because Japanese people themselves did not distinguish between は/ぱ in writing, and this made it more difficult for the missionaries to learn Japanese. That's the story I remember, at least.

Incidentally, changing a P sound to an F or otherwise F-like sound is a fairly common sound change across languages. That's why it's "father" in English but "padre" in Spanish, and that's also why the word "philosophy" is spelled with P's, too.

So our situation is now that we have a sound which is pronounced as F in most situations, and as P in a handful of places where the old pronunciation sort of fossilized. This is when we encounter another sound change, which is that often times between vowels, the F sound would become more V-like — which is to say that the vibration of the vocal folds from the immediately preceding and following vowels started to "bleed into" the F sound, that the vibration would stop too late or start too early relative to the movement of the lips, and this gave the F sound this more V like quality. And due to the acoustic similarity of this V-like pronunciation to the Japanese W sound, it ended up being conflated with the W sound and merging with it. But the old spelling stuck, which led to a situation prior to the postwar spelling reform, where は{行|ぎょう} and わ{行|ぎょう} could both be used to represent the W sound, depending entirely on the historical pronunciation of the word.

And indeed, the は particle, and for that matter the へ particle, were often reduced/slurred in such a way that they were basically treated like the ending of the previous word, and so these words were in fact often affected by this F-to-W sound change despite nominally being their own separate words which on their own wouldn't be affected. And this happened so often that は and へ ended up being pronounced as wa and we always. A similar shift in pronunciation happened to a lot of English-language function words that we spell with TH — the magic word in linguistics is "sandhi". So this is why "thy" and "thigh" are not pronounced the same, for instance.

And yeah, another sound change ended up merging the syllables wi, we, and wo with i, e, and o, so this is why へ is today pronounced as e rather than we. And then when the postwar spelling reform rolled in, it was decided that は/へ/を were particles used so frequently that they should just be left alone despite their historical spelling; otherwise, をゐゑ were respelled as おいえ, and every はひふへほ pronounced with a W sound was respelled as わいうえお, leaving the particle は as the absolute last and only remaining example of は{行|ぎょう} being read with a W sound. This spelling reform is incidentally also why there are no Japanese verbs ending in ふ, and why the Japanese verbs ending in う have わ as their {未然形|みぜんけい} rather than あ: the Japanese verbs ending in う historically ended in ふ prior to the spelling reforms.

But yeah. Not too long after the F-to-W sound change, most of the remaining examples of the Japanese F sound went through a different sound change, as the lips became less and less rounded, which gradually changed the blowing-out-a-candle F-like sound to a more simple exhaling H-like sound. This sound change was blocked whenever the Japanese F sound was immediately followed by a U, because that vowel also involves rounded lips, so that sort of reinforced the rounded lips of the F sound. And that's why はひふへほ is ha-hi-fu-he-ho instead of ha-hi-hu-he-ho. Badabing badaboom!

※ Note: the H sound before the vowel I ended up being palatalized, similarly to how we say the H in "huge". So while Hepburn romanization spells ひ as hi, the pronunciation of the H is a little different from the H in はへほ.

All this being said, is this actually useful information? Honestly, probably not super useful for most people, no. It might come a bit in handy if you ever try learning Chinese, Korean, Vietnamese, or Uchinaaguchi, though, because memorizing sound changes can help you identify, remember, or even to some extent predict the forms of the shared vocabulary between these languages. Otherwise equipping yourself with some knowledge of other historical Japanese sound changes or knowledge of phonetics can help make sense of some other oddities you will inevitably or potentially run into while learning Japanese, like why the volitional form of verbs might be described as sticking ～う to the end of a verb's {未然形|みぜんけい} form even though it just plainly isn't (spoiler: it was prior to the spelling reforms!); and why especially older people might say the particle が as "nga", and why {東北弁|とうほくべん} is Like That; why words might change their last vowel or first consonant when used in compounds and why the {濁点|だくてん} turns those specific consonants into those specific other consonants; why i/u are so often silent; why pitch accent patterns include the particles after a word; why the word です sounds almost like the English word "this" at 1:20 in the song ｢アイドル｣ by YOASOBI; and all sorts of other fun things like that.

Not that one couldn't learn through simple memorization and exposure, but I just think it's fun to know, and I think that having actual scientific or historical explanations helps the new information stick.

I'm not sure there's an answer beyond "because sometimes 'は' is pronounced 'wa'."

Kindof like asking "why is 'c' sometimes pronounced like an 's' like in 'cistern.'"

"こんにちは" is another word in which "は" is pronounced "wa".

I'd imagine there are probably historical reasons, but I don't know them. I suspect the cases where "は" is pronounced "wa" are pretty rare and just have to be learned by wrote.

It's WA when it's used as a particle and HA when not used as a particle. The Japanese government attempted to standardize WA sounds to わ after WWII, and was mostly successful, but the は particle stuck around, seemingly due to inertia. Lots of languages have little oddities in pronunciation that aren't reflected in spelling, or vice versa. Where do the British get the F in lieutenant?