It is idiom - but its meaning is pretty straight-forward (although exaggerated!) 'Fit' here means ready. Ready to burst means so full that you feel like you are going to explode. It could also be used for (say) a radiator hose in a car that has started to split.

"To be fit to (verb)" means you are ready to, or about to do something. It doesn't have a common use nowadays, apart from in expressions such as "fit to burst". You've eaten so much that you think you might burst, like a balloon.