Ten karton jest cięższy niż tamten, bo są w nim książki.

Because karton is a masculine singular noun in Polish, so the demonstrative has to match it:

• ten = masculine
• ta = feminine
• to = neuter

So:

• ten karton = this box/carton
• ta książka = this book
• to pudełko = this little box

The same agreement applies to tamten.

They are demonstratives:

• ten = this
• tamten = that

So the contrast is:

• ten karton = this box
• tamten = that one / that box

In this sentence, Polish is comparing two boxes: this one and that one.

Because Polish often leaves out a noun when it is already clear from context.

So:

• Ten karton jest cięższy niż tamten.

literally means:

• This box is heavier than that one.

The word tamten already implies tamten karton. English does the same with that one.

Because Polish does not have articles like a and the.

Polish usually relies on:

• context
• word order
• demonstratives like ten, tamten

So ten does not just mean the. It is stronger, more like this.
If you simply wanted The box is heavy, Polish would usually just say:

• Karton jest ciężki.

Not always exactly.

In this sentence, karton most naturally means a cardboard box or carton/packing box.

That is a little broader than English carton, which often suggests things like:

• a milk carton
• an egg carton

In Polish, karton can very naturally refer to a plain cardboard box used for packing or storing things.

Cięższy is the comparative form of ciężki (heavy), so it means heavier.

Base adjective:

• ciężki = heavy

Comparative:

• cięższy = heavier

It also agrees with karton, which is masculine singular, so the masculine singular comparative form is used.

Because with this adjective, Polish normally uses the simple comparative:

• ciężki → cięższy

That is the normal, natural way to say heavier.

Using bardziej ciężki would sound unnatural in most everyday situations.
Polish does sometimes use bardziej + adjective, but mainly when:

• there is no simple comparative,
• or the speaker wants a special stylistic effect.

Here, cięższy is definitely the standard choice.

Niż means than.

So:

• cięższy niż tamten = heavier than that one

Polish also often uses od after comparatives:

• Ten karton jest cięższy od tamtego.

Both are correct, but the grammar changes a little:

• after niż: tamten
• after od: tamtego (genitive form)

So both patterns are common:

• cięższy niż tamten
• cięższy od tamtego

Because the verbs agree with different subjects.

• Ten karton jest cięższy...
  Subject: ten karton = singular
  So: jest = is

• ...bo są w nim książki.
  Subject: książki = plural
  So: są = are

Even though English might say because there are books in it, Polish still uses normal plural agreement with książki.

Nim means him/it after a preposition. Here it means in it.

The base pronoun is:

• on = he / it (for masculine nouns)

After the preposition w (in), the form changes:

• w nim = in him / in it

Here it refers back to karton, so:

• w nim = in it / inside it

This is just the normal Polish pronoun form used after prepositions.

Here w expresses location (in), so it takes the locative case.

That is why you get:

• w nim = in it

This is different from motion into something, where Polish may use a different structure.

So in this sentence:

• w nim książki = books in it
• not movement, just location

Polish word order is more flexible than English.

The sentence:

• bo są w nim książki

is very natural and has an existential feel, similar to English:

• because there are books in it

You could also say:

• bo książki są w nim
• bo w nim są książki

but the emphasis would shift a little.

The original version sounds natural because it presents the books as the explanation for why the box is heavier.

Yes. Bo means because and is very common in everyday Polish.

So:

• bo są w nim książki = because there are books in it

You could also say:

• ponieważ są w nim książki

That sounds a bit more formal or written.

Very roughly:

• bo = everyday, common, conversational
• ponieważ = more formal, more careful style

In normal speech, bo is usually the most natural choice.