Han blev mere seriøs, da vi begyndte at tale om økonomi.

Because blive often means to become.

So Han blev mere seriøs means He became more serious.

If you said Han var mere seriøs, that would mean He was more serious, which describes a state, not a change.
Here, the sentence is about a change in his attitude when the topic changed.

They are both in the simple past.

• blev = past tense of blive
• begyndte = past tense of begynde

So the whole sentence describes something that happened in the past.

Danish often forms the comparative with mere + adjective, especially with longer adjectives and many loanwords.

So:

• seriøs = serious
• mere seriøs = more serious

This is very natural Danish. In practice, mere seriøs is the safe and common form to learn.

Because begynde is followed by an infinitive when you say what someone started doing.

• begynde at tale = to start talking

This is similar to English started to talk.

So:

• vi begyndte at tale = we started to talk

Because the verb tale om means to talk about.

• tale = talk/speak
• om = about

So:

• tale om politik = talk about politics
• tale om penge = talk about money
• tale om økonomi = talk about economics/finance/the economy

You usually need om after tale when the meaning is talk about something.

Da is used for a specific event in the past.

Here, the sentence refers to one past moment:

• da vi begyndte at tale om økonomi = when we started talking about economics/finance

By contrast, når is usually used for:

• repeated situations
• general truths
• future situations

Compare:

• Da vi begyndte at tale om økonomi, blev han mere seriøs.
  = one past occasion

• Når vi taler om økonomi, bliver han mere seriøs.
  = whenever we talk about economics/finance, he gets more serious

Because da introduces a subordinate clause, and subordinate clauses normally keep subject + verb order in Danish.

So you get:

• da vi begyndte
  not
• da begyndte vi

This is an important difference from main clauses, where Danish often uses verb-second word order.

Then the main clause must still follow Danish verb-second word order.

So you can say:

• Han blev mere seriøs, da vi begyndte at tale om økonomi.
• Da vi begyndte at tale om økonomi, blev han mere seriøs.

Notice the second version:

• Da vi begyndte at tale om økonomi, blev han ...
• not Da vi begyndte at tale om økonomi, han blev ...

When something other than the subject comes first, the verb usually comes before the subject in the main clause.

Because økonomi here is being used as a general topic, like economics, finance, or the economy in a broad sense.

So om økonomi means about economics/finance/economic matters.

If you said økonomien, that would usually sound more specific:

• økonomi = economics/finance as a subject
• økonomien = the economy

Both can be possible in different contexts, but om økonomi is very natural when talking about a topic of conversation.

Yes, very natural.

It means he grew more serious or became more serious in tone, expression, or attitude.

Danish often uses blive + adjective for this kind of change:

• blive glad = become happy
• blive vred = become angry
• blive stille = become quiet
• blive mere seriøs = become more serious

A rough guide:

• han ≈ han
• blev ≈ something like bleu / blaiw depending on accent, with a soft Danish vowel
• mere ≈ MEE-ruh
• seriøs ≈ seh-ree-URS or seh-ree-OS with stress near the end
• begyndte ≈ beh-GYN-duh
• tale ≈ TAY-luh
• økonomi ≈ ur-ko-no-MEE

The exact sounds are quite different from English, especially the vowels, but those approximations can help you recognize the words.

Yes. Depending on context, økonomi can mean:

• economics
• finance
• money matters
• the economy

So possible English translations include:

• He became more serious when we started talking about economics.
• He got more serious when we started talking about finance.
• He became more serious when we started talking about money matters.

The best choice depends on context, but the Danish grammar stays the same.