Banka je bliže mojoj kući nego restoran.

Here nego means than.

So:

• Banka je bliže mojoj kući = The bank is closer to my house
• nego restoran = than the restaurant

It introduces the second part of the comparison.

Je is the 3rd person singular present form of biti (to be), so here it means is.

• Banka je... = The bank is...

It works just like the English verb is in this sentence.

Because blizu means near/close, while bliže is the comparative form and means closer.

Compare:

• Banka je blizu moje kuće. = The bank is near my house.
• Banka je bliže mojoj kući nego restoran. = The bank is closer to my house than the restaurant.

So if the English meaning is closer, Serbian needs bliže.

In this kind of spatial expression, Serbian very often uses bliže as a predicate meaning closer/nearer. It behaves more like an invariable comparative form than like a regular adjective that has to match banka.

So Banka je bliže... is a normal way to say The bank is closer...

For a learner, the safest thing is to remember this pattern:

• blizu = near
• bliže = closer
• najbliže = closest

Because mojoj kući is in the dative case, and here it means to my house.

English uses the preposition to:

• closer to my house

Serbian often expresses that idea with the dative instead of a separate word like to.

So:

• moja kuća = my house (nominative, basic form)
• mojoj kući = to my house (dative)

Both words change:

• moja → mojoj
• kuća → kući

Here restoran is understood as the subject of an omitted clause, so it stays in the nominative.

The sentence is basically a shortened version of:

• Banka je bliže mojoj kući nego što je restoran.

That means:

• The bank is closer to my house than the restaurant is.

Since restoran is understood as the restaurant is, it remains in its basic subject form.

Because Serbian often leaves out repeated words when the meaning is clear.

So:

• Banka je bliže mojoj kući nego restoran.

is a shortened version of:

• Banka je bliže mojoj kući nego što je restoran.

English does this too sometimes:

• The bank is closer to my house than the restaurant.

Even though the full meaning is really:

• ...than the restaurant is.

Yes. You can say:

• Banka je bliže mojoj kući nego što je restoran.

That is more explicit and makes the comparison structure very clear.

The shorter version:

• Banka je bliže mojoj kući nego restoran.

is also natural, because the missing part is easy to understand from context.

Because Serbian does not have articles like English the and a/an.

So:

• banka can mean a bank or the bank
• restoran can mean a restaurant or the restaurant

Which one is meant depends on context. In your sentence, the meaning is understood from the situation, not from an article.

Yes. Serbian word order is more flexible than English word order.

The version you have is neutral and clear:

• Banka je bliže mojoj kući nego restoran.

But Serbian can move parts around for emphasis. Even so, beginners should usually stick to the neutral order first, because it is the easiest to understand and produce correctly.