Из-за шума в вагоне я едва слышу объявление.

Here из-за means because of.

So Из-за шума = because of the noise.

A useful extra note: из-за can also literally mean from behind in other contexts, for example из-за дома = from behind the house. In this sentence, though, it clearly has the causal meaning because of.

Because из-за requires the genitive case.

• шум = nominative singular
• шума = genitive singular

So:

• из-за шума = because of the noise

This is very common in Russian: certain prepositions always force a particular case.

Because в with the meaning in / inside / at a place usually takes the prepositional case.

• вагон = carriage / railcar
• в вагоне = in the carriage

So the form вагоне is the prepositional singular of вагон.

Compare:

• в вагоне = in the carriage
• в вагон = into the carriage

The first is location, the second is motion toward.

Вагон usually means a railway car / train carriage / railcar.

In this sentence, в вагоне most naturally means in the train car or in the carriage. Depending on context, it could also be a subway/metro car.

Едва here means hardly or barely.

So:

• я едва слышу объявление = I can barely hear the announcement

It suggests the speaker can hear it a little, but with difficulty.

Russian distinguishes between hearing and listening:

• слышать = to hear
• слушать = to listen

Here the idea is not that the speaker is actively listening to the announcement, but that they are barely able to hear it because of the noise.

So:

• Я едва слышу объявление = I can barely hear the announcement
• Я слушаю объявление would mean I am listening to the announcement

Слышу is the 1st person singular present tense of слышать:

• я слышу = I hear
• ты слышишь = you hear
• он/она слышит = he/she hears

So я едва слышу объявление literally means I barely hear the announcement.

In natural English, this is often translated as I can barely hear the announcement.

Because объявление is the direct object of слышу, so it is in the accusative case.

However, объявление is a neuter inanimate noun, and for neuter inanimate nouns the accusative singular looks the same as the nominative singular.

So:

• nominative: объявление
• accusative: объявление

That is why the form does not change.

Объявление can mean announcement, notice, or ad, depending on context.

In this sentence, because we are in вагоне, it most naturally means a public announcement, such as a train or metro announcement over the speaker system.

Russian word order is fairly flexible, and this order is natural because it puts the reason/background first:

• Из-за шума в вагоне = because of the noise in the carriage
• я едва слышу объявление = I can barely hear the announcement

Starting with Из-за шума в вагоне emphasizes the cause of the problem.

Other word orders are possible, for example:

• Я едва слышу объявление из-за шума в вагоне.

This means basically the same thing, but the focus is slightly different.

In this sentence, в вагоне most naturally goes with шума:

• из-за шума в вагоне = because of the noise in the carriage

So it tells us where the noise is.

In practice, the whole opening phrase gives the setting, so a listener understands both that the noise is in the carriage and that the speaker is there too.

Слышать is an imperfective verb, and it is the normal choice here because the sentence describes an ongoing present situation:

• I am barely able to hear the announcement
• not a completed single action

Russian often uses the imperfective for states, ongoing perception, and general present-tense situations.

Yes, Russian often omits subject pronouns when the verb form already makes the subject clear.

So you could also say:

• Из-за шума в вагоне едва слышу объявление.

Because слышу already means I hear, я is not strictly necessary.

Including я can add a bit of clarity or emphasis.