По прилёте таможенница попросила меня открыть чемодан и показать, что лежит внутри.

По прилёте is a set expression meaning upon arrival or after arriving.

A native English speaker often expects после прилёта, and that is also possible, but по прилёте is a very natural formal/written-style Russian expression. It belongs to a pattern like:

• по приезде = upon arriving
• по окончании = upon completion
• по возвращении = upon returning

So по прилёте is best learned as a fixed time expression.

Прилёте is in the prepositional case.

The noun is прилёт (arrival). Its prepositional singular form is прилёте.

Here it appears after по in a special temporal expression: по прилёте = upon arrival.

This is one of those places where Russian case usage is best learned as part of the whole phrase rather than by trying to force a simple one-word translation.

Таможенница means a female customs officer.

The masculine form is таможенник. Since the officer in the sentence is female, Russian uses the feminine noun таможенница.

This is very common in Russian: past-tense verbs and many nouns reflect gender clearly.

Попросила is the past tense, feminine singular form of попросить (to ask/request).

It agrees with таможенница, which is feminine:

• таможенник попросил = the male customs officer asked
• таможенница попросила = the female customs officer asked

So the -а ending tells you the subject is feminine.

Меня is the accusative case of я.

In the pattern попросить кого-то сделать что-то, the person being asked goes in the accusative:

• попросить меня = to ask me
• попросить его = to ask him
• попросить нас = to ask us

So:

• таможенница попросила меня открыть чемодан
  = the customs officer asked me to open the suitcase

Using мне would suggest a different structure, not the one used here.

Russian commonly uses this structure:

попросить кого-то + infinitive

So:

• попросила меня открыть = asked me to open
• попросила меня показать = asked me to show

This works very much like English asked someone to do something.

They are perfective because the customs officer asked for specific completed actions:

• открыть чемодан = to open the suitcase
• показать = to show

The request is not about an ongoing process or repeated activity. It is about doing those actions once, successfully, in that situation.

Compare:

• открыть = open (once, completed)

• открывать = be opening / open repeatedly / open in general

• показать = show (once, completed)
• показывать = be showing / show repeatedly / show in general

So perfective is the natural choice here.

Чемодан is the direct object of открыть, so it is in the accusative case.

But because чемодан is an inanimate masculine noun, its accusative singular looks exactly like the nominative singular:

• nominative: чемодан
• accusative: чемодан

So the case has changed grammatically, but the form stays the same.

Because Russian often avoids repeating something when it is already clear.

The full meaning is:

• попросила меня открыть чемодан и (меня) показать, что лежит внутри

But repeating меня would sound unnecessary. Russian, like English, often leaves out repeated elements when the meaning is obvious.

So the sentence naturally means:

• she asked me to open the suitcase and to show what was inside

Because что лежит внутри is a subordinate clause.

It depends on показать:

• показать, что лежит внутри
  = to show what is lying inside

Russian normally puts a comma before a subordinate clause introduced by что.

Here что means what and introduces the clause что лежит внутри.

So:

• показать, что лежит внутри
  = to show what is inside

This is not the same as что meaning that in every context. In this sentence, it functions more like what.

Лежит внутри is very natural when talking about the contents of a suitcase.

Literally, лежит means lies/is lying. Russian often uses лежать for objects that are physically somewhere, especially when imagining them resting in a place.

So:

• что лежит внутри = what is lying inside / what is inside

Находится внутри is possible, but it sounds more neutral, formal, or less vivid. For a suitcase, лежит внутри sounds very normal.

Because the grammatical subject of the clause is что.

In Russian, что here is grammatically singular neuter, so the verb is singular:

• что лежит внутри = what is inside

Even if the actual contents are multiple things, the structure still uses singular лежит because it agrees with что.

Here внутри is being used as an adverb, meaning inside.

So:

• что лежит внутри = what is lying inside

Russian can also use внутри as a preposition with the genitive:

• внутри чемодана = inside the suitcase

But in this sentence, the place is understood from context, so just внутри is enough.

Yes. Russian word order is flexible, although the original sentence sounds very natural.

For example, you could also say:

• Таможенница по прилёте попросила меня открыть чемодан и показать, что лежит внутри.

That still means the same thing.

The original version puts по прилёте first to set the scene immediately: upon arrival...

So the sentence starts with the time/context, then gives the main action. That is a very common Russian pattern.

Yes, a little.

По прилёте is perfectly normal Russian, but it sounds somewhat more formal, written, or narrative than a very conversational alternative like:

• когда я прилетел(а) = when I arrived
• после прилёта = after the arrival / after arriving

So learners should recognize по прилёте as common and useful, but also know that in casual speech people often choose simpler alternatives.