Languages are messy. They borrow words from each other, have irregular verbs and their spelling systems can sometimes do with a bit of work. Writing is a technology. It is something that we use to convert the sounds we make when speaking into characters that can be put down on the page (or screen in the case of computers).
Before dictionaries and standardised spelling came along people just wrote how they spoke. This was fine as long as the writing you were reading was written by someone with the same dialect as yours. Does your dialect drop R's? Then you can work out that "ka" meant an automobile. It was generally possible to work out what people meant.
But if you read a letter from elsewhere the differences between your own dialect and their dialect might make deciphering their letter a lot more difficult. Major standardisation of English spelling started happening around the time the printing press was introduced into England in 1477 by the printer William Caxton.
If you wanted to write things down in a standardised way and no such standard yet exists, then you need to make a standard. Caxton had to decide on what dialectal words to use but also what spelling to use. As printing spread, it took these new standardised spellings with them. Eventually people started getting used to the idea that a word should have one spelling.
So the words "busy" and "bury" ended up with their current spelling and even though it didn't really make sense to some speakers of the language, people just got used to it.
Ok, but why the U's? The reason I brought up dialects and standardisation is that they play an important role in this story. Why would people spell "busy" when it is pronounced "bisy"? Because some people did pronounce it like that. We call the English spoken at the time "Middle English". It was spoken from about 1100 to about 1500.
There were 5 dialects of Middle English: Northern, Southern, Kentish, East Midlands and West Midlands. The word "busy" is descended from the Old English word "bysiġ". The "y" in the Old English word is pronounced like the "ü" in German. It sounds like the "ee" in "free" but with rounded lips. In IPA it is /y/.
The "y" in Old English eventually started being pronounced like the "i" in "pit" in the East Midlands dialect. This dialect becomes important later as it is the dialect of London and Standard English is based on the dialect of this area. People who grew up in the East Midlands pronounced "busy" as "bisy" and spelled it as "bisy" because this was a bit before standardisation of spelling.
But around this time trade around the country started to increase and a lot of trade came through London. Traders from all over the country came through London and brought their pronunciations and their spellings with them. The "y" in Old English developed differently in the dialects of the West Midlands and the south. The "y" started being pronounced as a "u" (like in "tune" or "June") and they wrote their version of the word as "busy". In IPA this would be /busi/.
Standardisation didn't happen over night and spelling was still quite flexible. People might have used one or the other. Sometimes they used one that didn't really reflect their pronunciation but the differences weren't so big that people couldn't work it out. Dialects and spelling started to mix. But at the same time spelling started being standardised. It is almost like tree sap slowly flowing down a tree and trapping a mosquito.
Both spellings were in common use. It's like the language was wearing different outfits depending on where someone was from or what writings they had encountered. But bit by bit that sap flowed over the tree and this particular word became trapped like a mosquito in amber. As time went on and London became more influential and its dialect starting spreading around the country. At the same time English spelling became more standardised. Bit by bit the language settled down to a particular spelling of each word.
Another word that this happened to was "bury". This one too came from an Old English word containing "y". In this case it was "byrġan". It was spelled with a "u" too, just like "busy", because the "y" changed to a "u" sound in the West Midlands and the south.
But we pronounce "bury" like "berry", so what happened here? It is because of how it was pronounced in one of the dialects of Middle English, in this case the dialect of Kent. In this dialect, the Old English "y" sound eventually became an "e" sound so it would have been written as "berien" by these speakers. But like with "busy", the spelling of "bury" reflects the pronunciation from a different dialect.
If all the dialects had mixed sufficiently that a common spelling emerged before standardisation then the two words that are the stars of this article would probably have more sensible spellings. But society was changing, trade was increasing and people were moving around just at a time when the language was being standardised.
Spelling can sometimes be really boring but in this case it's a fascinating snapshot of a very interesting time in the history of the English language.