Dziś opiekunka odbiera córkę z przedszkola.

Because córkę is in the accusative case, which is the case usually used for the direct object of the verb.

The base form is córka = daughter.
In this sentence, the daughter is the person being picked up, so Polish changes the form:

• córka → nominative, dictionary form
• córkę → accusative

This is very common with feminine nouns ending in -a:

• kobieta → kobietę
• mamа → mamę
• córka → córkę

The preposition z meaning from/out of requires the genitive case.

So:

• przedszkole = nominative, dictionary form
• z przedszkola = genitive after z

This pattern is very common:

• z domu = from the house/home
• ze szkoły = from school
• z biura = from the office
• z przedszkola = from kindergarten

So the ending changes because the preposition controls the case.

Because opiekunka is the subject of the sentence, and subjects normally appear in the nominative case.

Here:

• opiekunka = the nanny / female caregiver
• she is the person who performs the action of odbiera

So the basic form stays:

• opiekunka odbiera... = the nanny picks up...

If you changed the grammatical role, the form could change too. For example:

• Widzę opiekunkę. = I see the nanny.
  Here opiekunkę is accusative because it is the object.

Here odbiera means picks up, in the sense of going somewhere and taking a person away from there.

So odbiera córkę z przedszkola means:

• she picks up the daughter from kindergarten

Be careful, because odbierać / odebrać can have several meanings depending on context, such as:

• pick up a child
• collect something
• receive something
• answer a phone call
• take away in some contexts

In this sentence, because of córkę z przedszkola, the meaning is clearly pick up.

Odbiera is present tense, so it means something like:

• is picking up
• picks up

In this sentence, Dziś gives a time frame of today, so in natural English it may often be translated with future meaning, such as Today the nanny is picking up the daughter from kindergarten. But grammatically in Polish, the verb form itself is present.

Compare:

• odbiera = present, imperfective
• odbierze = future, perfective

So:

• Dziś opiekunka odbiera córkę z przedszkola.
  Today the nanny is picking up / picks up the daughter from kindergarten.

• Dziś opiekunka odbierze córkę z przedszkola.
  Today the nanny will pick up the daughter from kindergarten.

Both can be possible in real life, but they express slightly different aspect and time meaning.

Yes, it could go somewhere else. Polish word order is quite flexible.

Dziś is at the beginning because it sets the time right away and sounds very natural. But these are also possible:

• Opiekunka dziś odbiera córkę z przedszkola.
• Opiekunka odbiera dziś córkę z przedszkola.
• Córkę z przedszkola dziś odbiera opiekunka.

The differences are mostly about emphasis, not basic meaning.

Beginning with Dziś is a very common way to frame the whole sentence: as for today...

Yes, but it is small.

Both mean today. In most situations, they are interchangeable.

• dziś = slightly shorter, often felt as a bit more concise
• dzisiaj = full form, very common in speech too

So you could also say:

• Dzisiaj opiekunka odbiera córkę z przedszkola.

The meaning stays the same.

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

So:

• opiekunka can mean a nanny or the nanny
• córkę can mean a daughter or the daughter

Which meaning is intended depends on context.

In this sentence, English will usually need the:

• Today the nanny picks up the daughter from kindergarten

But Polish simply leaves articles out.

The ending -ka often marks a feminine noun, and opiekunka is the feminine form of opiekun.

So:

• opiekun = male caregiver / guardian
• opiekunka = female caregiver / nanny / babysitter, depending on context

In this sentence, opiekunka specifically tells you the person is female.

Usually from kindergarten is the most natural English translation here.

Polish z often covers meanings that English expresses as from, off, or out of, depending on the noun and situation.

So here:

• z przedszkola = from kindergarten

You could say out of kindergarten in a very literal sense, but it sounds less natural in ordinary English for this context.

It is imperfective.

The verb pair is:

• odbierać = imperfective
• odebrać = perfective

Here, odbiera is the 3rd person singular present form of odbierać.

Imperfective is used for things like:

• ongoing actions
• repeated/habitual actions
• actions seen as a process
• present tense

That is why odbiera works naturally in a sentence like this. The perfective partner odebrać does not have a normal present meaning; its present forms usually refer to the future.

It can suggest either a planned event today or part of a routine, depending on context.

Because odbiera is imperfective present, Polish can use it for:

• a current or scheduled action: Today the nanny is picking up the daughter
• a repeated action in a framed situation: Today, as usual, the nanny picks up the daughter

Without more context, the sentence most naturally sounds like a statement about what is happening today.

A helpful rough guide is:

• ó sounds like Polish u
• ę is a nasal vowel, but before some consonants it is often pronounced in a simplified way

So córkę is roughly like TSOOR-keh with a nasal element on the last vowel, though English cannot represent it perfectly.

A few pronunciation points:

• c in Polish sounds like ts
• ó sounds like oo in food
• rk is pronounced clearly
• final ę is often less strongly nasal than learners expect

Similarly:

• przedszkola has the cluster prz-, which can be hard for beginners
• dziś ends with a soft ś, somewhat like a very soft sh

Not in normal correct Polish.

You should use the proper Polish noun przedszkole and then the correct case form after z:

• z przedszkola

If you borrow a foreign word, it still usually has to fit Polish grammar, but for a learner, it is much better to learn the standard Polish word and its case forms.

So the correct phrase here is:

• córkę z przedszkola