I found a translation of the well-known "studies at an English university" text (as long as the first and last letter are correct, the rest of the letters can be jumbled up and you can still read it) in Russian:

По рзелульаттам илссеовадний одонго анлигйсокго
унвиертисета, не иеемт занчнеия, в кокам пряокде
рсапожолены бкувы в солве. Галвоне, чотбы преавя и
пслоендяя бквуы блыи на мсете. осатьлыне бкувы
мгоут селдовтаь в плоонм бсепордяке, все-рвано ткест
чтаитсея без побрелм. Пичрионй эгото ялвятеся то,
что мы не чиатем кдаужю бкуву по отдльенотси, а все
солво цликеом.

I showed it to Sveta and she could, indeed, read it fairly quickly. (However, it seems that this version has largely preserved the order of consonants and of vowels in words, just mingling the two sets of letters a bit differently; this most certainly contributes to the legibility.)

Can someone tell me what that text should properly be? I can read part of the text, but I don't have enough vocabulary to recognise many of the words and figure out what they should be.

...A hunt for sources turned up this article talking about the phenomenon and this one from LanguageHat. And apparently Snopes has an entry on the topic, with links.