そうになる: reaching the brink of an event
Distinguish the near occurrence in そうになった from actual completion and a simple prediction.
A state just before something happens
Adding そうになる to a verb's ます-stem describes reaching a state in which the action seems about to occur. 転びます becomes 転びそうになる, which can describe almost falling.
階段で足を滑らせて、転びそうになりました。 means that the speaker slipped on the stairs and almost fell. The slipping did happen, but translating 転びそうになりました as actually falling would change the outcome.
The timing of そうだ and そうになった
泣きそうです。 describes being on the verge of tears now. 話を聞いて、泣きそうになりました。 looks back at the onset of that reaction: hearing the story brought the speaker close to crying.
This そう differs from そうだ for relaying something heard. 転びそうだ describes the appearance of being about to fall; 転んだそうだ reports having heard that someone fell. Check both the preceding verb form and the source of the information.
Do not decide what happened afterward
転びそうになった normally reports a moment when someone almost fell but did not. Unlike simply saying 転んだ, it focuses on the danger just before the event rather than the completed event itself.
With an emotional reaction such as 泣きそうになった, the expression likewise does not itself establish actual crying. A later clause may say that the person eventually cried, so do not add a claim that they never cried afterward either.
In the sentence ‘階段で足を滑らせて、転びそうになりました。’, what happened, and what almost happened?
Show the answer and why
The speaker's foot slipped, and the speaker almost fell. Replacing 転びそうになりました with 'fell' removes the distinction between approaching an event and actually experiencing it.