Jeg blev hjemme i går aftes, fordi jeg var meget træt.

Because blive can mean not only become, but also stay/remain in some contexts.

• Jeg blev hjemme = I stayed home
• Jeg var hjemme = I was at home

So blev hjemme suggests the action or decision of staying in, while var hjemme only tells you where the person was.

This is a very common Danish distinction:

• hjem = motion/direction, home
  • Jeg går hjem = I’m going home
• hjemme = location, at home
  • Jeg er hjemme = I’m at home

In your sentence, the speaker is located at home, not moving toward home, so hjemme is correct.

i går aftes means yesterday evening or last night.

It is made up of:

• i går = yesterday
• aftes = in the evening / last evening in a past-time sense

So together it clearly places the action in the evening of the previous day.

Yes, very often.

Both can mean last night / yesterday evening.

• Jeg blev hjemme i aftes
• Jeg blev hjemme i går aftes

i går aftes is a bit more explicit, because it literally says yesterday evening.
i aftes is very common in everyday Danish.

Because fordi introduces a subordinate clause, and in standard Danish subordinate clauses do not use main-clause word order.

So you get:

• fordi jeg var meget træt

not

• fordi var jeg meget træt

A useful contrast:

• Main clause: Jeg blev hjemme
• Subordinate clause: fordi jeg var meget træt

In standard written Danish, this subject-before-verb order after fordi is what learners should use.

Both are in the simple past.

• blev = past tense of blive
• var = past tense of være

That fits the time expression i går aftes, since the sentence is talking about a finished event in the past.

Because træt is being used as a predicative adjective after var.

After være in this kind of sentence, Danish uses the normal adjective form:

• jeg var træt = I was tired

With a plural subject, you would usually see trætte:

• vi var meget trætte = we were very tired

So jeg var meget træt is exactly what you would expect.

Because fordi jeg var meget træt is a subordinate clause, and many Danish writers put a comma before such clauses.

So this is very normal:

• Jeg blev hjemme i går aftes, fordi jeg var meget træt.

You may also see:

• Jeg blev hjemme i går aftes fordi jeg var meget træt.

Both can occur, depending on the comma style being used. So don’t be surprised if you see Danish written both ways.

Yes. Danish often moves time expressions to the front for emphasis.

You can say:

• I går aftes blev jeg hjemme, fordi jeg var meget træt.

Notice what happens then: the verb comes before the subject in the main clause:

• I går aftes blev jeg hjemme
• not I går aftes jeg blev hjemme

That is the normal verb-second pattern in Danish main clauses.

Yes, that is a good idea.

blive hjemme is a very common expression meaning stay home.

You can learn it as a useful unit:

• Jeg bliver hjemme = I’m staying home / I’ll stay home
• Jeg blev hjemme = I stayed home
• Bliv hjemme! = Stay home!

That will help you avoid trying to translate it too literally as became home, which is not how it works in Danish.

That is true of a lot of Danish words.

In most standard pronunciation, jeg sounds roughly like yai to English ears, not like it is spelled.

So the sentence begins approximately like:

• Jeg blev hjemme ≈ yai blew/home-ish...

But Danish pronunciation varies a lot, and the exact sounds are difficult to represent with English spelling. The main thing to know as a learner is:

• jeg is usually not pronounced like a hard yeg
• it is much closer to yai

If you want, I can also break down the pronunciation of the whole sentence word by word.