So what exactly is a fitt? With its double terminal stop, it’s a rather strange-looking word for English, isn’t it? The word refers to a section of medieval Germanic verse, basically the equivalent of the Latinate canto. More on that in a moment, but first, to clarify the meaning and source of the word:
Like all the longer Old English poems, Beowulf is divided into sectional divisions that were in all likelihood denoted by the term fitt ‘fitt,’ pl. *fitta. [Footnote: This is to be deduced from the Latin preface to the Heliand, which states that the author ‘omne opus per vitteas distinxit, quas nos lectiones vel sententias possumus appellare’ (‘divided the whole work into fitts, which we can call “readings” or “passages”) [...] 
As you can see by the footnote above, fitts weren’t limited to verse in Old English but were also present in the literature of other Germanic languages, as in the Old Saxon Heliand. Fitts were sometimes delineated by punctuation, sometimes with numerals, sometimes by elaborately decorative initial capitals. Other times, one has to deduce where one fitt ends and another commences. Fitts often ended with general statements of summary, reflection, or wisdom , as for example in the advice concluding fitt 35 of Beowulf — sibb æfre ne mæg wiht onwendan þám ðe wel þenceð (“kinship can never aught pervert in him who rightly thinks”).
Coming back to canto, as many of you will know, this too is a word denoting a section in a long poem. It comes to us through Italian from the Latin cantare “to sing”. A fitt, it transpires, seems to show precisely the same derivation, but on the Germanic branch of the Indo-European Stammbaum. There, the immediate source would appear to be Old English fittan “to sing”. It appears to me not to be related to fitt “strife, stuggle”, though some have claimed it is. OE fitt “poem, song” is attested only four times, and it was not until Chaucer that the word fitt was used in the same sense as canto (in The Tale of Sir Thopas, there spelled “fit”).
But the more common OE word for “to sing” was singan; so where might fittan have come from? I think there are two possibilities, both metaphorical — and I’m not sure which one I favor:
(I) A phonologically similar Old High German word, fizza “yarn, skein, hank (of thread)”, suggests that singing could have been related to the spinning of a yarn (= a tale). Cognate to this are Middle High German vitzen and Old Norse fitja “to web, knit, weave”. 
(II) Alternatively, the Old Norse fet “pace, step”, secondarily “foot (as of a poem), a part of a poem” — one must assume related to fótr “foot” — suggests the possibility of singing while walking, measuring out a poem (or other oral literature) at or by the pace of walking, etc.
But these theories are really just speculation. Various dictionaries suggest the possibility of one or the other (or both), sometimes with certainty, sometimes not. In the end, I think we must admit that one can only go back so far before the vagaries of linguistic change become lost in the haze of history. Arguing backward, we just make the best fitt we can.
 Klaeber, Friedrich. Klaeber’s Beowulf and the Fight at Finnsburg. 4th ed. by R.D. Fulk, et al. Toronto: University of Toronto Press, 2008, p. xxxiii.
 Loc. cit., and xxxv. [The translation of the passage that follows is Benjamin Thorpe’s.]
 R.D. Fulk, one of the editors of the most recent edition of the Klaeber Beowulf, has an article on this subject: “The origin of the numbered sections in Beowulf and in other Old English poems.” Anglo-Saxon England 35 (2006): 91–109. There, inter alia, he talks about the possibility of a derivation from the OHG fizza.