Fit to VERB in this case† is a construction with the approximate meaning "to the point of VERBing". "I cried fit to bust" means "I cried so hard I almost burst", and you might encountered this in "I laughed fit to bust" or "He was mad fit to bust", meaning "I laughed so hard I almost burst" or "He was so angry he seemed like he would burst".
Bust here is a colloquial variant of burst. Other verbs may also occur in the construction.
Historically the construction derives from the adjective fit, "suitable, prepared" with the extended sense of "inclined" or "disposed", and it was at one time used predicatively; but the adjectival sense has long since evaporated from the construction, which is to the best of my knowledge now used
only adverbially,‡ and only in colloquial and dialect contexts. I wouldn't bother with any more granular syntactical analysis: this is a fixed construction.
† Fit to VERB is also used with its ordinary adjectival sense "suitable for VERBing" or "suitable to be VERBen"; that use is standard, fit to employ in any register.
‡ TRomano reminds us of fit to be tied used predicatively, so I've struck my hasty generalization. But I think the construction is adverbial far more often than adjectival.