Az átvételi ponton átveszem a küldeményt, és megmutatom az értesítést.

Because they are in the definite conjugation.

In Hungarian, if the verb has a definite, specific object, the verb often takes definite endings. Here the objects are:

• a küldeményt = the parcel / shipment
• az értesítést = the notification

Both are definite, so the verbs are:

• átveszem = I pick it up / receive it
• megmutatom = I show it

Compare:

• Átveszek egy küldeményt. = I pick up a parcel.
  → indefinite conjugation
• Átveszem a küldeményt. = I pick up the parcel.
  → definite conjugation

So the verb ending is reacting to the definiteness of the object.

That -t is the accusative ending, which marks the direct object in Hungarian.

In this sentence, the things being acted on are:

• a küldeményt = the parcel/shipment
• az értesítést = the notification

So:

• küldemény → küldeményt
• értesítés → értesítést

English mostly marks direct objects by word order, but Hungarian often marks them directly with -t.

Ponton is pont + -on, and here it means at the pickup point.

The suffix -on/-en/-ön is often called the superessive case. Its basic meaning is on, but in many expressions it is used more idiomatically as at.

So:

• az átvételi ponton = at the pickup point

Why not pontban?

• -ban/-ben usually means in / inside
• -on/-en/-ön can mean on or idiomatically at

With service points, counters, stops, and similar locations, Hungarian often prefers this kind of at the point/place wording.

Átvételi is an adjective meaning something like for pickup / related to receiving.

It comes from:

• átvétel = pickup, receipt, taking over
• -i = an adjective-forming suffix

So:

• átvételi pont = pickup point / collection point

This is a very common Hungarian pattern:

• fizetés = payment → fizetési mód = payment method
• belépés = entry → belépési kód = entry code

So átvételi pont is literally something like pickup-related point.

They are verbal prefixes, and they often change the meaning of the base verb.

• vesz = take, buy

• átvesz = take over, receive, pick up

• mutat = show, point out
• megmutat = show, demonstrate

So the prefixes are not just decorative; they are part of the vocabulary item.

They also often give the verb a more completed or bounded feel, though the exact effect depends on the verb.

Another important point: in Hungarian, prefixes can move away from the verb in some sentence types.

For example:

• Átveszem a küldeményt. = neutral
• A küldeményt veszem át. = emphasis on the parcel
• Nem mutatom meg. = I do not show it

So in your sentence, the prefixes stay attached because the sentence is in a neutral declarative order.

Because Hungarian usually does not need subject pronouns.

The verb endings already show who the subject is:

• átveszem = I pick up
• megmutatom = I show

So én is normally omitted unless you want emphasis or contrast.

For example:

• Én veszem át a küldeményt. = I am the one picking up the parcel.
• Én megmutatom az értesítést, ő pedig aláír. = I show the notification, and he/she signs.

In a neutral sentence, leaving out én is the most natural choice.

Hungarian uses the definite article very regularly, often more regularly than English.

Here, az átvételi ponton means at the pickup point, referring to a specific one relevant in the situation.

Also, Hungarian has two forms of the:

• a before a consonant
• az before a vowel

Since átvételi begins with a vowel sound, the article is az:

• az átvételi ponton
• az értesítést

But:

• a küldeményt
• a ponton

Because it is understood from the verb form.

In:

• Az átvételi ponton átveszem a küldeményt, és megmutatom az értesítést.

both verbs are first person singular, so Hungarian does not need to repeat én.

Repeating the subject is possible, but usually only for emphasis:

• ..., és én megmutatom az értesítést.

That would sound more contrastive, as if you are stressing that I am the one who shows it.

In a normal neutral sentence, the repeated subject is omitted.

Hungarian word order is fairly flexible, but changes in order usually affect emphasis or focus, not just style.

Your sentence is a natural, neutral version:

• Az átvételi ponton átveszem a küldeményt, és megmutatom az értesítést.

This presents the location first, then the two actions.

You could reorder parts of it, but the emphasis changes. For example:

• A küldeményt az átvételi ponton veszem át.
  Emphasis is stronger on the parcel
• Az értesítést is megmutatom.
  Emphasis on the notification too

So the original word order is not the only possible one, but it is a very natural default if you are simply describing what happens.