Мне стало нехорошо в метро, поэтому я вышел на следующей станции.

Russian often uses an impersonal pattern for physical or emotional states: мне холодно, ему грустно, ей плохо. The person who experiences the state goes in the dative case, so мне means to me / for me.

So мне стало нехорошо literally works more like it became unwell for me, but the natural English meaning is I started to feel unwell.
Я стал нехорошо would not work here. If you used я стал..., you would need an adjective or another complement, for example я стал злым.

Стало is the past tense of стать, meaning to become.

In this kind of impersonal sentence, there is no normal grammatical subject, so Russian uses the neuter singular past form by default: стало. That is why it is not стал or стала.

The pattern мне стало + state word is very common:

• мне стало плохо
• мне стало грустно
• мне стало страшно

It usually means I began to feel...

Стало emphasizes a change of state.

• Мне стало нехорошо = I started to feel unwell
• Мне было нехорошо = I was feeling unwell

So стало is about the moment or onset, while было just describes the condition.

Yes, you could. In fact, мне стало плохо is probably more common in everyday speech.

The difference is mostly one of tone:

• плохо = more direct, very common
• нехорошо = a little softer, sometimes more euphemistic

In this sentence, нехорошо means physically unwell, not not nice or morally bad.

Here в метро means in the metro / on the subway, meaning the speaker was physically there.

Russian usually uses:

• в метро = in the subway, inside it, while riding
• на метро = by metro, as a means of transport

For example:

• Мне стало нехорошо в метро = I felt unwell in the subway
• Я езжу на метро = I go by metro

Also, метро is an indeclinable noun, so its form stays метро after prepositions.

Поэтому means therefore, so, or that’s why. It introduces the result of the first clause.

So the logic is:

• Мне стало нехорошо в метро = cause
• поэтому я вышел на следующей станции = result

Russian normally uses a comma before поэтому when it connects two clauses like this.

It is different from потому что:

• поэтому = so / therefore
• потому что = because

Compare:

• Мне стало нехорошо, поэтому я вышел.
• Я вышел, потому что мне стало нехорошо.

Вышел is the past tense of the perfective verb выйти.

Two important things are shown here:

1. Aspect:
   The perfective is used because this was one completed action: he got off once, at the next station.

2. Gender:
   вышел is masculine singular, so the speaker is male.
   A female speaker would say вышла.

So:

• я вышел = a man says I got off
• я вышла = a woman says I got off

With stations and stops, Russian normally says:

• на станции = at the station
• на остановке = at the stop

Here the phrase tells you where the getting-off happened, so на is used in a location sense. That means Russian uses the prepositional case here:

• на следующей станции

The adjective следующей agrees with станции in gender, number, and case.

So this phrase literally means at the next station.

Because выйти из метро usually means to leave the metro system/building.

But вышел на следующей станции means got off at the next station. The focus is on leaving the train at that stop, not necessarily leaving the whole metro entirely.

In Russian, when the vehicle is obvious from context, you can often just say вышел without repeating exactly what you got out of.