Jo større rygsækken er, jo flere bøger kan jeg tage med til universitetet.

It means the ... the ..., just like in English comparisons:

• Jo større rygsækken er, jo flere bøger kan jeg tage med ...
• The bigger the backpack is, the more books I can bring ...

This structure is very common in Danish for proportional comparisons.

A few more examples:

• Jo mere jeg læser, jo bedre forstår jeg.
  = The more I read, the better I understand.
• Jo tidligere du kommer, jo bedre.
  = The earlier you arrive, the better.

So jo introduces each half of the comparison.

Because Danish uses the comparative form here.

• stor = big
• større = bigger
• størst = biggest

Since the sentence compares sizes, Danish needs the comparative:

• Jo større rygsækken er ...
  = The bigger the backpack is ...

This is exactly like English using bigger instead of big.

Because bøger is a countable noun.

In Danish:

• flere = more, used with countable things
• mere = more, used with uncountable things

So:

• flere bøger = more books
• mere vand = more water
• mere tid = more time

Since you can count books, Danish uses flere.

Rygsækken means the backpack, while en rygsæk means a backpack.

Danish usually adds the definite article to the end of the noun:

• en rygsæk = a backpack
• rygsækken = the backpack

Here, the sentence is talking about the backpack in a general but definite sense: the bigger the backpack is...

This matches English, which also normally says the bigger the backpack is, not a bigger backpack is in this kind of structure.

Because the first clause is structured like a subordinate clause, where the normal order is:

subject + verb

So:

• rygsækken er = the backpack is

In Danish main clauses, the verb often comes in second position, but after jo in this kind of comparative construction, the clause behaves like a subordinate clause, so the subject comes before the verb.

That is why you get:

• Jo større rygsækken er ...

not:

• Jo større er rygsækken ...

Because the second half behaves like a main clause, and Danish main clauses follow the V2 rule: the finite verb goes in the second position.

Here the clause begins with:

• jo flere bøger

That whole phrase takes the first position. Then the finite verb must come next:

• jo flere bøger kan jeg tage med ...

So the order is:

1. jo flere bøger
2. kan
3. jeg
4. tage med ...

This is why it is kan jeg tage, not jeg kan tage, in this sentence.

Tage med means take/bring along.

In this sentence:

• kan jeg tage med til universitetet = I can take/bring along to the university

So the full idea is:

• the more books I can take with me to the university

The particle med is important here. Without it, tage just means take, but tage med often means take along/bring along.

Examples:

• Jeg tager min computer med.
  = I’m bringing my computer along.
• Kan du tage bogen med?
  = Can you bring the book?

Not necessarily. In Danish, tage med can already imply take along/bring along, and the destination makes the meaning clear.

So:

• kan jeg tage med til universitetet

naturally means something like:

• can I bring/take along to the university

If you wanted to make with me explicit, you could say:

• kan jeg tage med mig til universitetet

But that is often unnecessary when the meaning is obvious from context.

Because til expresses movement toward a destination:

• til universitetet = to the university

By contrast:

• på universitetet = at the university / at university

So:

• Jeg tager bøger med til universitetet.
  = I bring books to the university.
• Jeg læser på universitetet.
  = I study at the university.

In your sentence, the speaker is talking about bringing books to the university, so til is the correct preposition.

Yes, in standard Danish writing, the comma between the two jo clauses is normal and expected:

• Jo større rygsækken er, jo flere bøger kan jeg tage med til universitetet.

The comma helps separate the two linked parts of the comparison.

Danish comma rules can be a little different from English, but in a sentence like this, the comma is standard.

Yes, absolutely. This is a very productive pattern in Danish.

Template:

• Jo + comparative + clause, jo + comparative + clause

Examples:

• Jo ældre jeg bliver, jo klogere bliver jeg.
  = The older I get, the wiser I become.
• Jo mere du øver dig, jo bedre spiller du.
  = The more you practice, the better you play.
• Jo længere vi venter, jo senere kommer vi hjem.
  = The longer we wait, the later we get home.

So your sentence is a very useful model to learn from.

In this Danish sentence, English could use either take or bring, depending on viewpoint:

• The bigger the backpack is, the more books I can take to the university.
• The bigger the backpack is, the more books I can bring to the university.

Danish tage med often covers both ideas more flexibly than English does. English makes a stronger distinction between take and bring, but Danish is often less strict in everyday use here.

So if you were translating it, either English version could work depending on context.