I was learning about how to identify the passé simple yesterday and wondered why the passé simple of « être » is « fut ». It's very dissimilar, so much so that I mistook it for « faire ».
The verb être is built from a mixture of two Latin verbs, esse (to be) and stāre (to stay), moreover, the latin esse had some tenses built from proto-Indo-European roots, like precisely fuī (which shares the same root as the English to be).
This background explains the very irregular forms depending on the persons and the tenses.
You are lucky not studying Spanish, where the situation is even more complex.
See also this thesis about suppletive verbs (like être) in Romance languages.
Être is from two Latin stems, esse and stāre, but this doesn't actually cover the problem because both the être and fut forms are from only one of those two stems.
That stem, esse, was itself a combination of two Proto-Italic stems, according to Wiktionary.
The present stem is from Proto-Italic *ezom, from Proto-Indo-European *h₁ésti (“I am, I exist”).
The perfect stem is from Proto-Italic *fūai, from Proto-Indo-European *bʰúHt (“to become, be”).
Naturally, the first root is the source of être and probably also soi- as in soit and ser- as in sera (though if another linguist on here can verify or deny that, please do!). The second root is the source of fut (and, incidentally, is related to futur). Meanwhile, the stems beginning with ét- as in était are from that altogether different Latin root stāre.
That these separate lexemes combined to form one verb in Latin is certainly interesting. As jlliagre points out, English has this too: be and was are ultimately from different sources. (What may surprise you most is that the be is ultimately from the same root that spawned Latin esse root and was to the one that spawned fut...! That said, when we are dealing with Proto-Indo-European, we must recall that we are speculating thousands of years back and are trying to explain commonalities between languages, so this sort of extreme etymology shouldn't be relied on too heavily.)
Note that the two-stem stage and the one-stem stage are different languages, separated in usage by long periods of time. Perhaps the fusion happened as children sometimes heard the one, sometimes the other, and the meanings were so close that they ended up misreading the import. They would unconsciously reason along these lines: "Oh, Mommy says it like that when it happened yesterday and like this when it's happening today." As Chomsky and many linguists of his school have observed, the data children get about their language is so unbelievably sparse compared to the complexity of the subject matter that such things happen all the time and are only sometimes noticed and corrected by parents (and only sometimes do children respond to explicit grammatical correction!). Such silent reinterpretations are the engine of a lot of language change, such as the transition that will eventually made from the main verb "to want" to a modal indicating the future tense. Or the amusing story of how the English word "apron" ultimately comes from the French "napperon": the English phrase "a napron" was parsed as "an apron" by enough members of a few generations.
However the fusion happened, though, what can be generalized is the following observation: the most common words in a language are usually the ones with most bizarre behaviour. "To be" is an everyday word. So is "to go", where we see three roots in French: aller, ir- as in ira, and va. "Women" represents a very strangely pronunciation for the letter "o". And so on. (I could give examples from Hebrew and Spanish as well.) This is somewhat surprising given the above explanation; you'd think that the most common words would be the ones children would hear often enough not to be mistaken about them. But conversely, being heard often enough can also cement a mistake. :)
It would be fascinating to read a source giving more detail on exactly how and when those two Proto-Italic roots fused into one verb in Latin — jlliagre's sources look very promising, and are making me rethink ser- and soit — but hopefully this gives you an idea of what's involved.
Isn't it the same in Latin already: sum → fui, est → fuit? I would says that's the origin of our passé simple. sum/être is a highly irregular verb, hence the radical change in form for some tenses, most probably because it is used a lot.