Min veninde siger, at hendes rygsæk er for stor til at have med i taxaen.

Veninde means female friend. The corresponding general/male word is ven.

It usually does not mean a romantic girlfriend. For a boyfriend/girlfriend in the romantic sense, Danish normally uses kæreste. So min veninde is simply my female friend.

After a possessive word like min, din, hendes, vores, etc., Danish uses the basic noun form, not the suffixed definite form.

So:

• veninde = female friend
• veninden = the female friend
• min veninde = my female friend

And:

• rygsæk = backpack
• rygsækken = the backpack
• hendes rygsæk = her backpack

A possessive already makes the noun definite enough, so you do not add -en or -et as well.

Because sin/sit/sine only refers back to the subject of the same clause.

In the subordinate clause at hendes rygsæk er for stor ..., the grammatical subject is hendes rygsæk, not the friend herself. That means Danish does not use sin here.

Compare:

• Hun tager sin rygsæk med.
  Here hun is the subject of the clause, so sin is correct.

• Min veninde siger, at hendes rygsæk er for stor.
  In this clause, the friend is not the subject, so hendes is used.

So even if the backpack belongs to the friend, hendes is the correct form in this sentence.

Because adjectives in Danish agree with the noun’s gender and number.

Rygsæk is a common-gender singular noun: en rygsæk.
So the adjective is stor.

Compare:

• en rygsæk er stor
• et hus er stort
• to rygsække er store

So er for stor is correct because the subject is one common-gender noun.

They are doing two different jobs.

1. The first at in siger, at ... means that.
   It introduces a subordinate clause.

2. The second at in til at have med is the infinitive marker, like English to in to have.

So:

• at = that
• at = to

They look the same, but they are not the same kind of word here.

Here for means too.

So:

• stor = big
• for stor = too big

This is a very common Danish pattern:

for + adjective + til at + infinitive

Examples:

• for tung til at bære = too heavy to carry
• for dyr til at købe = too expensive to buy
• for stor til at have med = too big to have with you / bring along

So the for here is not the English preposition for.

It follows the pattern:

for + adjective + til at + verb

This means:

too + adjective + to + verb

So:

• for stor = too big
• til at have med = to have with you / to bring along

Together, for stor til at have med means too big to have with you.

This structure is extremely common in Danish and worth memorizing as a whole pattern.

Have med literally means have with. In natural English, that often comes out as have with you, carry with you, or bring along.

In this sentence, it suggests the backpack would be accompanying the person in the taxi.

You could often also use tage med, which means take along / bring. The difference is roughly this:

• tage med focuses more on the action of bringing something
• have med focuses more on something being with you in the situation

So have med is very natural here.

Because they express different things.

• med = with you / along
• i taxaen = in the taxi

So have med i taxaen means have it with you in the taxi.

If you removed med, the sentence would focus only on location, not on the idea of bringing the backpack along with you.

Taxaen is the definite form: the taxi.

Danish usually marks definiteness by adding a suffix to the noun:

• en taxa = a taxi
• taxaen = the taxi

In this sentence, taxaen suggests a specific taxi or taxi ride that the speaker has in mind.

A more general version could use en taxa, but taxaen sounds like a particular situation.

Because Danish normally uses i for being inside small vehicles such as cars and taxis.

So:

• i bilen = in the car
• i taxaen = in the taxi

That is the normal choice here.

Yes.

In at hendes rygsæk er for stor ..., the subject comes before the finite verb:

• hendes rygsæk = subject
• er = finite verb

That is normal Danish subordinate-clause word order.

A useful extra point: if you added an adverb like ikke, it would usually come before the verb in the subordinate clause:

• at hendes rygsæk ikke er for stor

So this sentence is a good example of standard Danish clause structure.