In this case, it means three quarters. However, note that “part” does not have any particular relationship with “quarter”.
The same usage appears in The Lord of the Rings:
‘Behind us in the caves of the Deep are three parts of the folk of Westfold, old and young, children and women,’ said Gamling. ‘But great store of food, and many beasts and their fodder, have also been gathered there.’
When I asked a similar question on English Language & Usage, based on this passage, the answer from FumbleFingers pointed out such related usages as possession is nine parts of the law and kindness is ninety-nine parts of what passes for wisdom. The phrasing in all these terms simply implies that there is one remaining part. If three parts of the goblin warriors of the North died, one part remains; if three parts of the folk of Westfold are in the caves, one part is elsewhere; if possession is nine parts of the law, then everything else is one part; if kindness is ninty-nine parts of what passes for wisdom; then everything else is one part. And so on.
This is a slightly old-fashioned usage. Somehow, I inferred the four, but wasn’t sure where I’d got it from, which is why I asked my question on EL&U rather than here on SciFi & Fantasy. The answer, though, is the same. However many parts are mentioned in the phrase, the whole consists of one part more than that.
As FumbleFingers also mentioned, three quarters is an easily understood fraction, much easier for the human mind to grasp and visualise than, for example, four fifths. As such, three parts is likely to be a more common element of these phrases than four parts would be. That’s probably why my brain filled in the gap and understood that three parts meant three quarters, but didn’t recognise the pattern involved. In the phrases using higher numbers, such as nine and ninety-nine, that might just be an expression of the common pattern of dividing things into tenths and hundredths.
Given the prevalence of the duodecimal system in old weights and measures, one might expect eleven parts to be widely used, but I’ve never come across it. This might make an interesting question for EL&U.