Čím víc knih si moje dcera chce půjčit, tím větší tašku potřebuje.

It is the Czech equivalent of the more ..., the ...er ....

So this sentence is built like:

• Čím víc knih ... = the more books ...
• tím větší tašku ... = the bigger a bag ...

This is a very common Czech pattern for showing that one thing changes together with another.

A good way to remember it is:

• čím introduces the changing condition
• tím introduces the result

Examples:

• Čím jsem starší, tím jsem klidnější. = The older I am, the calmer I am.
• Čím víc spíš, tím líp se cítíš. = The more you sleep, the better you feel.

Because Czech treats the Čím ..., tím ... structure as two linked clauses, and they are normally separated by a comma.

So:

• Čím víc knih si moje dcera chce půjčit, tím větší tašku potřebuje.

That comma is standard and expected in writing.

After words of quantity such as víc, méně, hodně, málo, Czech usually puts the noun in the genitive plural.

So:

• knihy = nominative plural
• knih = genitive plural

That is why you get:

• víc knih = more books

Compare:

• hodně lidí = many people
• málo času = little time
• víc peněz = more money

This is a very important pattern in Czech.

Yes. Both are correct:

• Čím víc knih ...
• Čím více knih ...

The difference is mostly style:

• víc is very common and natural in everyday speech
• více sounds a bit more formal or written

So in normal conversation, víc is extremely common.

Because půjčit si means to borrow, literally something like to lend to oneself / get lent for oneself.

This is a key Czech distinction:

• půjčit něco někomu = to lend something to someone
• půjčit si něco = to borrow something

So here:

• moje dcera si chce půjčit knihy = my daughter wants to borrow books

Without si, the meaning would change.

Because si is a clitic, and Czech clitics usually go in the second position of their clause.

In this clause, the opening chunk is:

• Čím víc knih

After that, the clitic si appears early:

• Čím víc knih si moje dcera chce půjčit

Even though si belongs in meaning with půjčit, its position follows Czech clitic rules, not English-style word order.

This is one of the trickiest things for English speakers: Czech word order is not based only on which words belong together logically.

Because půjčit si is perfective and refers to a completed borrowing act: she wants to borrow some books.

That fits this sentence well, because the idea is that she wants to borrow a certain number of books, and therefore needs a bigger bag.

If you used půjčovat si, that would sound more like repeated or habitual borrowing, depending on context.

So:

• chce si půjčit = wants to borrow
• chce si půjčovat = wants to be borrowing repeatedly / habitually

With chce + infinitive, Czech often uses the aspect that matches the intended meaning very closely.

Because tašku is the direct object of potřebuje, and direct objects are normally in the accusative.

So:

• taška = nominative singular
• tašku = accusative singular

The adjective must agree with the noun, so we get:

• větší tašku = a bigger bag

A detail that often confuses learners: větší looks the same in nominative and accusative feminine singular. That is normal for this comparative form. The noun tašku shows the case clearly.

Because the meaning is she needs a bigger bag, not she needs more bags.

The sentence is talking about one bag whose size changes depending on how many books she wants to borrow.

If the intended meaning were plural, Czech could use plural:

• potřebuje větší tašky = she needs bigger bags

But that is not what this sentence says.

Both are possible.

• moje dcera = my daughter
• má dcera = my daughter

In many everyday contexts, moje sounds very natural and common.
má is also correct, but can sound a bit shorter, sometimes slightly more literary or more emphatic depending on context and intonation.

For a learner, the safest thing is:

• both are correct
• moje is very common in normal speech

Yes. Czech word order is flexible, and different orders can sound natural depending on emphasis.

For example, you could also hear:

• Čím víc knih si chce moje dcera půjčit, tím větší tašku potřebuje.
• Moje dcera si chce půjčit čím víc knih, tím větší tašku potřebuje. — much less natural as a full sentence, because the correlative structure strongly prefers the standard pairing

The original sentence is very natural and clear.
When word order changes in Czech, the meaning often stays the same, but the focus or information structure changes.

One thing usually stays important, though: clitics like si still tend to keep their early position in the clause.