eki no kaidan de subette, korobisou ni narimasita.

の links two nouns.

• 駅 = station
• 階段 = stairs
• 駅の階段 = the station’s stairs or the stairs at the station

In natural English, this is usually just the stairs at the station or the station stairs.
So の is not doing anything complicated here; it is simply connecting station and stairs.

Here, で marks the place where an action happens.

So:

• 階段で滑って = slipped on the stairs
• not to the stairs
• not from the stairs

A common comparison:

• で = where an action takes place
• に = destination, time, or existence, depending on the sentence

In this sentence, the action 滑って happened on the stairs, so で is the natural particle.

The て-form is being used to connect what happened first to what happened next.

• 滑って = slipping / slipped, and then...
• 転びそうになりました = almost fell

So the sentence shows a sequence:

1. I slipped
2. As a result, I almost fell

The て-form often connects actions like and, but it can also suggest cause or result.
Here it feels like both:

• I slipped, and then...
• Because I slipped, I almost fell

These two verbs are related, but they are not the same.

• 滑る = to slip
• 転ぶ = to fall over / to tumble

So in this sentence:

• first, the person slipped
• then they were about to fall over

This makes the sentence very natural. Slipping does not always mean you actually fall.
That is why 滑って、転びそうになりました works so well: it describes losing footing and then nearly falling.

そう attached to a verb stem often means looks like it will... or is about to...

• 転ぶ = to fall over
• verb stem = 転び
• 転びそう = looks like someone will fall / seems about to fall

In this sentence, it is best understood as:

• about to fall
• likely to fall
• almost falling

So 転びそう by itself means something like it looked like I was going to fall.

Both are possible, but they are a little different.

• 転びそうでした = I looked like I was going to fall / It seemed I would fall
• 転びそうになりました = I ended up almost falling / I got into a state where I was about to fall

～そうになる is a very common pattern for almost doing something.

So:

• 転びそうになりました strongly suggests I came close to falling
• it emphasizes the moment when things nearly turned into a fall

That is why it is especially natural after 滑って.

The に is part of the pattern ～そうになる.

Here is the idea:

• 転びそう = a state of seeming about to fall
• なる = to become

So 転びそうになる literally means:

• to become in a state of seeming about to fall

In more natural English, that becomes:

• to almost fall
• to come close to falling

So the に is not random; it is the particle often used before なる when something changes into a state.

Because the whole event happened in the past.

• なります = present/future polite
• なりました = past polite

The speaker is describing something that already happened:

• they slipped
• they almost fell

So past tense is required.

Also, ました makes the sentence polite.
In casual speech, you might hear:

• 駅の階段で滑って、転びそうになった。

Japanese often leaves out the subject when it is clear from context.

So even though English usually needs I, Japanese often does not say it.

This sentence could mean:

• I slipped on the station stairs and almost fell
• or, depending on context, he/she slipped...

But in many real-life situations, if someone is telling their own story, the listener naturally understands I.

This is one of the biggest differences from English:
Japanese very often omits words that are already understood.

No, it is not grammatically necessary.

The comma is just there to make the sentence easier to read. It shows a pause between:

• 滑って
• 転びそうになりました

You will often see commas used this way in Japanese writing, especially when:

• the sentence is a little long
• the writer wants to make the flow clearer
• there is a natural pause

So the sentence without the comma is still fine:

• 駅の階段で滑って転びそうになりました。

No, the grammar strongly suggests they did not actually fall.

• 転ぶ = to fall
• 転びそうになる = to almost fall

So the person got close to falling, but did not fully fall over.

If they actually fell, you would expect something like:

• 駅の階段で滑って、転びました。
• 駅の階段で滑って、転んでしまいました。

That is an important nuance:

• 転びました = fell
• 転びそうになりました = almost fell

Yes. 転びかけました is also natural and means I almost fell.

But there is a nuance difference:

• 転びそうになりました = I got into a state where I was about to fall
• 転びかけました = I nearly started to fall / I almost fell

In everyday use, both can work here.
For a learner, ～そうになる is a very useful and common pattern to remember because it appears with many verbs:

• 死にそうになる = almost die
• 泣きそうになる = almost cry
• 忘れそうになる = almost forget

So this sentence is a good example of that pattern.