Na putu do aerodroma bila je velika gužva, ali je naš vozač stigao na vreme.

Because na putu is locative, and here it means on the way or en route.

The phrase na putu do + genitive is a very common Serbian pattern:

• put = road, way
• na putu = on the road / on the way
• na put would usually suggest motion onto a trip or journey, not location

So Na putu do aerodroma means On the way to the airport.

Because the preposition do requires the genitive case.

So:

• aerodrom = airport
• aerodroma = genitive singular

After do, Serbian uses genitive to mean to, up to, or as far as:

• do škole = to the school
• do grada = to the city
• do aerodroma = to the airport

Because bila agrees with gužva, and gužva is a feminine singular noun.

In the Serbian past tense, the participle agrees with the subject in gender and number:

• bio = masculine singular
• bila = feminine singular
• bilo = neuter singular
• bili / bile / bila = plural forms depending on gender

Here the thing that was is gužva, so the sentence uses bila.

Gužva literally means something like crowd, crush, congestion, or jam.

In this sentence, because the context is on the way to the airport, it most naturally means heavy traffic or a traffic jam.

So velika gužva here is not just a lot of people in general; it is more like:

• bad traffic
• serious congestion
• a big traffic jam

Serbian often uses gužva by itself when the context already makes it clear what kind of congestion is meant.

Because adjectives must agree with the noun they describe.

Here:

• gužva is feminine singular
• so the adjective also has to be feminine singular
• velik → velika

So:

• velika gužva = a big traffic jam / heavy congestion

You can compare:

• velik problem = a big problem (masculine)
• velika gužva = a big traffic jam (feminine)
• veliko selo = a big village (neuter)

Serbian word order is more flexible than English, but the short auxiliary je is a clitic, which means it is unstressed and tends to appear near the beginning of its clause.

In this sentence, bila je velika gužva is a natural and standard order.

A few important points:

• Serbian often moves elements around for rhythm, focus, or style
• the meaning usually stays the same
• clitics like je have placement rules, so they do not behave like normal full words

For a learner, the safest thing is simply to remember bila je here as a normal Serbian pattern.

Again, je is a clitic, so it likes to appear very early in the clause.

After conjunctions like ali (but), Serbian often places the clitic immediately after the conjunction:

• ali je naš vozač stigao na vreme

That is very natural Serbian.

You may also hear other word orders in real speech, but this one is standard and common. The main thing to understand is that je is an unstressed helper word, so it does not stay fixed in one English-like position.

Stigao is the past form of stići, which means to arrive or sometimes to make it / get there.

Here:

• vozač is masculine singular
• so the past participle is stigao (masculine singular)

This verb is also perfective, which means it presents the arrival as a completed event:

• he arrived
• he made it

That works very well in this sentence, because the point is that despite the traffic, the driver successfully arrived on time.

Na vreme is a fixed expression meaning on time or in time.

So:

• stigao na vreme = arrived on time

You should learn it as a set phrase. Even though it literally contains na, it does not mean physical movement onto time. It is just the normal idiomatic Serbian way to express punctuality.

Because Serbian, like English, often leaves out information that is already obvious from context.

The first clause already tells us the situation was on the way to the airport, so in the second clause it is easy to understand where the driver arrived:

• he arrived there
• meaning: at the airport / at the intended destination

So stigao na vreme is enough. Repeating na aerodrom would be possible, but it is not necessary.

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

So Serbian says:

• na putu do aerodroma
• velika gužva
• naš vozač

and context tells you whether something is definite or indefinite.

In this sentence:

• naš vozač is naturally definite because our already identifies him
• aerodroma is understood from context as the airport
• velika gužva is simply a big traffic jam / heavy congestion, depending on how you translate it into English